Things do exist just not in the way they appear to be existing.
Lacking inherent existency means that things/objects/phenomena are not set up by themselves. They are set up dependent on other factors. In other words they are conditioned.
That can be seen with e.g. a tree. A tree changes during the different seasons. Why does it change? It changes due to being set up dependent on other factors which again is set up dependent on other factors etc. That means that objects are in a constant flux. They do not stay constant or stable.
If a tree where inherent existent it would never change. It would stay green with flowers on it forever. If it truly were inherent existent then it would be uncaused. It would be completely defined by its own nature. Nothing could be removed or added to it since that would change its inherent existent nature. It would be both eternal and indestructible.
If we take a look at samsara the conditioned existence then we see that all physical and mental phenomena are caused by other factors. They are not stand-alone units. That does not mean that they are not real. It just means that they do not exist in the way they appear to be.
The Dalai Lama has written many great sections on Emptiness in his book "How to See Yourself As You Really Are". Here is a short quote on the existence of phenomena. The quote is from chapter 19: "Viewing Yourself As Like an Illusion", p. 187-188:
"A face in a mirror appears to be a face, but this image is not a real face in any way; it is from all viewpoints empty of being a face. Likewise, a magician can conjure up illusions that seem to be certain things, like a person in a box being skewered by a sword, but they are not at all established as those things. Similarly, phenomena such as bodies appear to be established from the objects' own side but are empty of being established that way and always have been.
It is not that phenomena are illusions; rather, they are like illusions. Even if a mirror image of your face is not really your face, the reflection is not utterly nonexistent. Through its appearance you can understand how your actual face looks. Similarly, although persons and things are empty of existing the way they appear to be established in their own right, they are not utterly nonexistent; they can act and be experienced.
Therefore, being like an illusion is not the same as appearing to exist but actually not existing, like the horns of a rabbit, which do not exist at all."
Here is another one with the title "Emptiness does not mean Nothingness". The quote is from chapter 5: "Appreciating the Reasoning of Dependent-Arising", p. 71-73:
"There is no question that persons and things exist; the question is how, or in what manner, they exist. When we consider a flower, for instance, and think, "This flower has a nice shape, nice color, and nice texture," it seems as if there is something concrete that possesses these qualities of shape, color, and texture. When we look into these qualities, as well as the parts of the flower, they seem to be qualities or parts of the flower, such as the color of the flower, the shape of the flower, the stem of the flower, and the petals of the flower - as if there is a flower that possesses these qualities or parts.
However, if the flower really exists the way it appears, we should be able to come up with something separate from all of these qualities and parts that is the flower. But we cannot. Such a flower is not found upon analysis, or through other scientific tools, even though previously it seemed so substantial, so findable. Because a flower has effects, it certainly exists, but when we search to find a flower existing in accordance with our ideas about it, that is not at all findable.
Something that truly exists from its own side should become more and more obvious when analyzed - it should be clearly found. But the opposite is the case. Nevertheless, this does not meant hat it does not exist, for it is effective - it creates effects. The fact that it is not found under analysis just indicates that it does not exists the way it appear to our senses and to our thoughts - that is, so concretely established with itself."
I would really recommend reading this book since it has a lot of insights about emptiness. I think sometimes its greatly beneficial to study another buddhists school's take on e.g. Anatta/Emptiness in order to "get it" cooked and prepared in another way. Of course the basic doctrine is the same but when reading about it in other terms and ways of expression that gives variety and nuances on the topic.
On purpose i did not go into the insight-meditational aspect of this profound teaching that emptiness is since you asked for an intellectual opinion. Hope this might be of some help.