You say it well. Although you focus on the, let's say, less popular part of spontaneity; burps and farts, I think very few people would support naturalness if it was just that :)
But you are right, although I had not thought about that, I was thinking more about how people live by consciously modifying our behavior to please certain people, or to seem good or better at something, to seem more capable or to be a better option, etc. Basically we lie without saying words, we just act in a way that we are not, pretending to be something, because lying is simply pretending to be something that you are not, and most of the time we don't act as we are, and therefore, we are not what we pretend.
Yes, freedom and nature go hand in hand, you cannot have one without the other.
I liked the quote, even though I didn't quite understand it, don't we live in our own modern caves? Maybe I missed something.
RE: Nature vs man