I have encountered Kh1 ideas in many variations of the Be3 MDP (once in a correspondence game), and you're right that it is a typical engine move. As I was analyzing the correspondence, I found all of these small little points:
1. There are no tricks along the a7-g1 diagonal.
2. White has Bg1 defensive ideas available, even meeting ...g3 with Bg1.
3. White can play Rg1, and even possibly g3.
4. Taking time out for a prophylactic move can sometimes force Black to show his intentions or make a move he didn't want to make without some commitment from White, e.g. the typical tension between Nb5 and ...g5 - White wants to play Nb5 quickly to generate kingside play, but it undefends the e-pawn and frequently allows ...g5-g4 without the preparatory ...h5. But waiting a move might force Black to commit to ...h5 anyway for lack of a better alternative.
I'm not sure it's actually better than an active move for White, but eventually I came to at least moderately respect the idea of Kh1 in the Be3 MDP, even if I ended up obtaining a winning position in my game.