I'll try to relate that to what the suttas say.
Enlightenment is inherently desirable
I'm not sure what "enlightenment" is -- i.e. what, in Buddhist doctrine, is translated as "enlightenment"?
Enlightenment in Buddhism [Wikipedia] suggests it's any of several terms:
- bodhi -- knowledge or wisdom, etymologically related to "awake"
- vimutti -- release (i.e. related to being freed)
- others -- paññā (insight), kenshho and satori (Japanese), nibbana, and Buddhahood
If I try to summarise Buddhism in one word, it's not "enlightenment" but rather "dukkha-nirodha" i.e. "cessation of suffering".
Self-Interested desires prevent us from attaining enlightenment
I'm not sure it's useful to use "self" in language.
In particular it's "wrong view" -- a thicket -- to say for example "self exists" and/or "self doesn't exist".
If you need to use that language then the ideal is "practice which benefits both self and others (AN 4.95)".
I find it clearer to remove "self" from the sentence, so instead of seeing something as "my suffering" just see it as "suffering".
So the sentence becomes, "desires prevent us from attaining enlightenment".
And in reply to that, Buddhism distinguishes between wholesome and unwholesome desires -- i.e. taṇhā (which is referenced in the Second Noble Truth), and chanda which is defined for example here or in a bit more detail here, the point being that chanda (one type of desire) isn't necessarily a bad thing.
You may be right though, that if you don't know what enlightenment is, and don't know how to go about it, and want it anyway, then that "wanting" and so on may be the unwholesome form of desire, i.e. an example of the kind of thing that actually causes suffering and delusion.
I'm not sure how items 3 and 4 in the OP were meant to add to the alleged paradox, so I won't address them directly -- they seem to be more of 2, i.e. talking about "self-interested" and "desire".
You might be interested in the Brahmana Sutta (SN 51.15) which answers someone's saying that "it's impossible that one could abandon desire by means of desire" (using the word chanda for desire). The answer is that you desire e.g. to go to the park (I suspect that "park" there is a reference to "forest monastery" i.e. where the monks are), and so you go (i.e. do that), and when you get there then the corresponding desire is allayed (i.e. paṭippassaddha which I think of as a nirodha again).
I guess that, in summary, nirodha isn't impossible. Maybe you have to recognise what it is that sustains (perpetuates) dukkha, and not attach to that.