Why are we checking the flop?

He might have a set? He also might have TT or JJ.

You realllllly have to call this as played. You have massively under-repped your hand postflop, villain doesn't have any 6's.
He's repping 99 or overpair. You beat most of this and if he shows up with 99 then w/e, just a cooler.

AA I don't worry about either as they'll be 99.5% of the time raising back preflop.

Looks like a standard bet bet bet line to me. As played I'm calling the river, GG if they have the last 6 in the deck, but I'm fully expecting to see over pairs in their range which are betting for value given your flop check.

I see absolutely no reason for not betting this flop, especially when there are draws. This 28/19 villain type can easily have 54s and many flush draws.

On this run out I would just bet all three streets.

As played you have to call the river with all the worse over pairs he can play this way.