[Nmoitmzr]

They can theorize, hypothesize, postulate, philosophize and pontificate all they want, but that inherent sense of "I" will still be there. I would go so far as to say that it it will always be there for everyone, regardless of what they do or don't do, or how "enlightened" they are.

The Buddha never told that "nothing is there" but that the self as a inherent entity is as illusory as to think that a rabbit has appeared suddenly in the hat of the magician.

[...]theorize, hypothesize, postulate, philosophize and pontificate[...]

Exactly, this are mental entanglements (of fabrications in the words of Buddha) that lead us to the idea of a inherent self, something that "is there". So, can we tell exactly where is such a "there"? That "there" is always changing, fading way and we tend to attach to it. If not, why stress? If not, why we change our mind all the time?

[...]this sense of "I" will always remain.

Exactly, it is a sense. It has no inherent substance. The quest is to know HOW, and just how, this sense comes to be so real for us.

So, experience is something that happens to consciousness.

Not exactly. Experience builds consciousness. Not only told by the Buddha but by all the theoretical edifice of Developmental Psychology that tells how consciousness is just a "by product" of experience and it is not just "given".

For example, in anthropology there is a reaserch field about how this thing called "consiusness" is socially build. Examples of feral children found in the wilderness show the kind of consciousness given by the experience of being reared by dogs, wolves or apes but not the kind of consiusness build up by human society. Once the have human social experiences they develop a kind of human social consiousness.