I disagree with just about all of your rationale but somehow still find myself arriving at the same (bolded) conclusion.
They're going to get xp through exploring the world and dealing with the challenges it throws at them. What I specifically and intentionally want, however, is a setup where those characters who do things and take risks are rewarded more than those who do not, and in a more concrete and measurable way than just DM fiat.
I don't care if characters in the same party are of different levels. That's just not a consideration, particularly seeing as my game has bespoke progression tables for each class and also includes things like level-draining undead.
As for "going where the right number of xp to level is", with one rare exception this doesn't come up as advancement in my game is very slow anyway. They just go where the adventure is (and the treasure, of course!). The rare exception is when a character is just a very few xp away from bumping, and in the fiction I've no real problem with the character feeling "that close" to needing to train again and going out to get those last few xp. (I once had a character get stuck for a while at 119,998 xp when he needed 120,001 to bump - where was a Goblin when I needed one!)
I also don't give out xp after every encounter or even after every session, unless I know someone's on the cusp of bumping. I give them out in batches instead, every half-dozen sessons or so usually, or at the end of any adventure just to close the books on it. (particularly if I know there's about to be some character turnover while in town)