Was it the Buddha's intention that all beings to be free from suffering i.e. enlightened?
As the Dalai Lama acknowledges, “Buddhism has evolved differently in different times and places and yet the essential Dharma remains the same. The Buddha’s prime concern was that all beings should find peace and freedom from suffering.
Did he say whether it was possible for everyone to be enlightened, so that there was no one left unenlightened? Would there then be no further (re)birth?
The Buddha does not have the intention for everyone to be enlightened.
However, the Buddha has compassion, which is his expression of alleviating the suffering of others. He is willing to teach everyone who is willing to listen to him.
So, the end goal of enlightening everyone is not his goal. However, his goal is to teach others who are willing to listen, and hope that they will also help others.
The teaching of others is his goal, and not the end goal of enlightening them. That's up to the people whom he teaches. It's in their hands, not his.
The Desanaa Sutta shows the 3 types of people he is willing to teach, to various degrees:
"Does not the Blessed One dwell in compassion for all living beings?"
"Indeed, headman, the Tathaagata does dwell in compassion for all living beings."
"Well then, Lord, does not the Blessed One teach Dhamma in full to some, but not so fully to others?"
"I will reply to this question, headman, with another. Answer as seems proper to you. What do you think? Suppose a peasant farmer has three fields, one excellent, one middling, and one poor, sandy, salty, with bad soil. Tell me: when the farmer wants to sow his seed, which field would he sow first: the excellent one, the middling one or the poor one that is sandy, salty and with bad soil?"
"Lord, the farmer who wanted to sow his seed would sow the excellent field first. Having done that, he would sow the middling field next, and the one that was poor, sandy, salty, with bad soil he might or might not sow. Why? Well it might do for cattle-food."
"Well, headman, that excellent field is like my monks and nuns. To them I teach the Dhamma which is lovely in its beginning, lovely in its middle and lovely in its ending, in spirit and in letter, I display to them the holy life, perfectly fulfilled and purified. Why? Because these people adhere to me as their island, their shelter, their resort, their refuge.
"The middling field is like my male and female lay-followers. To these too I teach the Dhamma which is lovely in its beginning, lovely in its middle and lovely in its ending, in spirit and in letter, I display to them the holy life, perfectly fulfilled and purified. Why? Because these people adhere to me as their island, their shelter, their resort, their refuge.
"The poor field that is sandy, salty and with bad soil is like my wandering recluses and Brahmans of other sects. To them I also teach the Dhamma which is lovely in its beginning, lovely in its middle and lovely in its ending, in spirit and in letter, I display to them the holy life, perfectly fulfilled and purified. Why? Because if they only understand a single phrase, it would long be for their profit."
The Kesi Sutta shows that, as an excellent teacher, he has various techniques to teach others, but he may give up on stubborn students:
"Kesi, I train a tamable person [sometimes] with gentleness, [sometimes] with harshness, [sometimes] with both gentleness & harshness.
"In using gentleness, [I teach:] 'Such is good bodily conduct. Such is the result of good bodily conduct. Such is good verbal conduct. Such is the result of good verbal conduct. Such is good mental conduct. Such is the result of good mental conduct. Such are the devas. Such are human beings.'
"In using harshness, [I teach:] 'Such is bodily misconduct. Such is the result of bodily misconduct. Such is verbal misconduct. Such is the result of verbal misconduct. Such is mental misconduct. Such is the result of mental misconduct. Such is hell. Such is the animal womb. Such the realm of the hungry shades.'
"In using gentleness & harshness, [I teach:] 'Such is good bodily conduct. Such is the result of good bodily conduct. Such is bodily misconduct. Such is the result of bodily misconduct. Such is good verbal conduct. Such is the result of good verbal conduct. Such is verbal misconduct. Such is the result of verbal misconduct. Such is good mental conduct. Such is the result of good mental conduct. Such is mental misconduct. Such is the result of mental misconduct. Such are the devas. Such are human beings. Such is hell. Such is the animal womb. Such the realm of the hungry shades.'"
"And if a tamable person doesn't submit either to a mild training or to a harsh training or to a mild & harsh training, what do you do?"
"If a tamable person doesn't submit either to a mild training or to a harsh training or to a mild & harsh training, then I kill him, Kesi."
"But it's not proper for our Blessed One to take life! And yet the Blessed One just said, 'I kill him, Kesi.'"
"It is true, Kesi, that it's not proper for a Tathagata to take life. But if a tamable person doesn't submit either to a mild training or to a harsh training or to a mild & harsh training, then the Tathagata doesn't regard him as being worth speaking to or admonishing. His knowledgeable fellows in the holy life don't regard him as being worth speaking to or admonishing. This is what it means to be totally destroyed in the Doctrine & Discipline, when the Tathagata doesn't regard one as being worth speaking to or admonishing, and one's knowledgeable fellows in the holy life don't regard one as being worth speaking to or admonishing."
The Maghadeva Sutta shows that the Buddha does not want his disciples to stop practising his teachings, and wants them to continue practising and teaching throughout the generations:
But now I have founded a good practice that does lead to disillusionment, fading away, cessation, peace, insight, awakening, and extinguishment. And what is that good practice? It is simply this noble eightfold path, that is: right view, right thought, right speech, right action, right livelihood, right effort, right mindfulness, and right immersion. This is the good practice I have now founded that leads to disillusionment, fading away, cessation, peace, insight, awakening, and extinguishment. Ānanda, I say to you: ‘You all should keep up this good practice that I have founded. Do not be my final men.’ Whatever generation is current when such good practice is broken, he is their final man. Ānanda, I say to you: ‘You all should keep up this good practice that I have founded. Do not be my final men.’”
Finally, it's up to people whether they want to become enlightened or not, according to the Mahaparinibbana Sutta:
"Therefore, Ananda, be islands unto yourselves, refuges unto yourselves, seeking no external refuge; with the Dhamma as your island, the Dhamma as your refuge, seeking no other refuge.
"And how, Ananda, is a bhikkhu an island unto himself, a refuge unto himself, seeking no external refuge; with the Dhamma as his island, the Dhamma as his refuge, seeking no other refuge?
"When he dwells contemplating the body in the body, earnestly, clearly comprehending, and mindfully, after having overcome desire and sorrow in regard to the world; when he dwells contemplating feelings in feelings, the mind in the mind, and mental objects in mental objects, earnestly, clearly comprehending, and mindfully, after having overcome desire and sorrow in regard to the world, then, truly, he is an island unto himself, a refuge unto himself, seeking no external refuge; having the Dhamma as his island, the Dhamma as his refuge, seeking no other refuge.
"Those bhikkhus of mine, Ananda, who now or after I am gone, abide as an island unto themselves, as a refuge unto themselves, seeking no other refuge; having the Dhamma as their island and refuge, seeking no other refuge: it is they who will become the highest, if they have the desire to learn."
Answer to the original question:
The Buddha doesn't have wantings per se. He has uprooted craving. He simply helps beings to achieve freedom from suffering out of compassion. Beings are infinite, so any thought of making everyone enlightened is absurd. The Buddha certainly had no such ambition.
Answer to later additions:
I do not follow Dalai Lama and the use of the word 'concern' is not appropriate as enlightened beings are not 'concerned' per se. But there's a big difference between saying what everyone should do and having an ambition that everyone will do it. Ex: It's fine to say "everyone should eat healthy food at all times". Because that's just good advice to all. But having that as an ambition is not realistic.
A Buddha does not want what is not possible.
59. Even so, on the rubbish heap of blinded mortals the disciple of the Supremely Enlightened One shines resplendent in wisdom.
174. Blind is the world; here only a few possess insight. Only a few, like birds escaping from the net, go to realms of bliss.
Dhammapada
Then, having understood Brahma's invitation, out of compassion for beings, I surveyed the world with the eye of an Awakened One. As I did so, I saw beings with little dust in their eyes and those with much, those with keen faculties and those with dull, those with good attributes and those with bad, those easy to teach and those hard, some of them seeing disgrace & danger in the other world.
'Open are the doors to the Deathless to those with ears. Let them show their conviction. Perceiving trouble, O Brahma, I did not tell people the refined, sublime Dhamma.'
MN 26
And, Master Gotama, when having directly known it, you teach the Dhamma to your disciples for the purification of beings, for the overcoming of sorrow & lamentation, for the disappearance of pain & distress, for the attainment of the right method, & for the realization of Unbinding, will all the cosmos be led [to release], or a half of it, or a third?
When this was said, the Blessed One was silent.
AN 10.95
I think that some Mahayana answers suggest that it is possible to participate in, to partake of, to benefit from the Buddha's enlightenment -- see for example here, or here.
Instead of writing "it's possible to participate" above, I thought of writing "it's possible for everyone or for anyone to participate" (or, "it's possible for anyone who wants to, to participate"). However, introducing the concept of "everyone" or "each person" may not conform to the doctrine of anatta or sunyata -- see also the distinction between lokuttara and lokiya.
And it's exactly that (i.e. asking about "everyone") might be a/the problem inherent in the question.
I mean, you ask whether "all beings" can be free from suffering i.e. enlightened -- but maybe it's identifying as "a being" that is (or is a cause of) the actual problem which is an antithesis of enlightenment -- see for example Vajira Sutta:
Why now do you assume 'a being'?
Mara, have you grasped a view?
It is possible, though, to infer other answers from the canon -- e.g. there's the story of the weaver's daughter, which is in the commentary associated with Dhp 174 quoted in Dhammadhatu's answer.
That story is quite a person-centred view (it says that this person became a stream winner before she died, that most of the villagers weren't so enlightened, that the father became an arahant afterwards) -- and (especially relevant to the question) it implies that the Buddha saw she had the capacity to become enlightened, and delayed his Dhamma talk specifically to wait for her.
Also note that, at least according to the Pali canon, there are different stages of enlightenment.
Go forth, O Bhikkhus, for the good of the many, for the happiness of the many, out of compassion for the world..
A Buddha is born via generating the uncontrived Mind of Enlightenment. This is an immense compassion and wish to free all sentient beings from suffering completely. It is such an uncommon mind that only Arya Bodhisattva's and fully enlightened Buddha's can even imagine what this mind would be like to experience.
As for the other answers... I think it is quite silly to say that the Buddha is not concerned or does not wish to help unlimited sentient beings or that this is somehow not realistic. I think this is not so subtle way of criticizing the Mahayana for being somehow unreasonable or unrealistic. This is rather ironic given that even the Pali tradition itself contains teachings on the Bodhisattva vehicle. Consider the Four Immeasurables which can be found in the Pali canon. What are the Four Immeasurables? They are a form of mind training for Bodhisattva's wishing to generate the Mind of Enlightenment:
These thoughts are said to be limitless precisely because sentient beings are limitless. To say that generating this mind is unrealistic in order to criticize the Mahayana is silly and unproductive. It is profoundly beneficial to train your mind to generate these uncontrived thoughts. Even more, one cannot become a Buddha without doing so! How silly to say that because the goal is impossible that generating this mind is not worthwhile!
The quintessential training in compassion is exemplified by Shantideva's Guide to the Bodhisattva Way of Life. It is essentially an expansion and elaboration of the very same attitudes and minds expressed by the Buddha in the Pali Canon's sutta's on the Four Immeasurables. Shantideva's guide is in my opinion one of the most beautiful and sacred religious texts on earth. I would happily dare anyone to read it completely and come away with a different opinion :)
Yes, as the Buddha states in the Diamond Sutra:
All living beings, whether born from eggs, from the womb, from moisture, or spontaneously; whether they have form or do not have form; whether they are aware or unaware, whether they are not aware or not unaware, all living beings will eventually be led by me to the final Nirvana, the final ending of the cycle of birth and death.
