To start off, i definitely wouldn't consider your river value bet thin at all. Paired board makes it less likely he has a 6, so your hand clearly looks the best given the specific scenario.
Your turn decision depends on his preflop 3 betting range. Obviously if he's only 3 betting Aces or Kings preflop, the turn is a easy fold (But you do have both the A and K blocker so I understand your thought process a bit). Also your flush draw could possibly be no good, so IMO the folds pretty standard if he has a nitty 3 betting range.
I definitely wouldn't be 4 betting 100% pre, but keep in mind you should have these hands in your 4 bet jam range. Nice river btw