I think it's an open question as to how engaged the public ever really is with politics. Quite a small number of people make quite a lot of noise and that always obscures how much the average person really cares about anything. I suspect that a not insignificant chunk of the "Leave" vote was a protest not at the EU but at politics in general, although it's hard to prove that of course so it's just a feeling I have -- and it would be pretty pathetic on my part to assume that all Remain voters went that way because they had carefully considered their vote and all the ramifications thereof.
Not sure how to expand the "different idea on how democracy might work" as you ask, I'm afraid. It feels fundamentally wrong to expect the losing side to keep quiet from now on. Beyond that it's just an aspiration that, whatever else the outcome, Brexit improves the way we do democracy in this country. So far it appears not to have, as Labour have just reverted to tearing themselves apart as usual and the Tories managed to avoid having a new leadership vote, although to be fair that was because Leadsom quit the race early rather than anything Theresa May did. After Brexit, I hope we'd look again at various questions related to the UK's democracy that would become far more urgent and important to resolve.