AA confusion - what now?

[arahant]

Live 1/2/5 blinds, 200 buy in. I'm dealt AdAh in early lp.
Table is mostly loose passive. I haven't been playing long enough to have much of an image at table.
My stack is 350, covers all but maybe 2 ep players. Most stacks below 200.
The player directly to my left raises to 20-30 regularly, and i believe (mistakenly, it turns out) that he has a tell for when he'll be raising.
4 limpers to me, and i limp believing that the player to my left will raise. Tragically, he too limps, as well as button and blinds. 9 to the flop, and i decide that barring a miracle, i'm more or less done.

Flop KT6d. All check to me. Thoughts?

I didn't think much about it at the table, but decided to push on the grounds that -
1) a pot bet leaves me needing to call a reraise from all but 2 players at table
2) with luck, i'll get called by a worse pair/draw feeling that i'm bullying or on a draw. (the push being the classic "i have a flush draw" move at this table).

[shootaa]

pot the flop, ure losing lots of value by pushing.

[sevenhigh]

only thing that calls a flop push is a flush. if its 1,2 and 9 people saw the flop the pot is only 18 a 20 dollar bet leaves u enough room to fold to an AI , IMO. at tables like this i find that people will often call with the king or q dimonds but not an AI.

[Bill King]

so you pushed right away? dont you think thats getting a little too overzealous? you could probably see the turn and river heads up or 3-way if you make a decent raise/bet. i just dont understand pushing like that.

[arahant]

Hehe...'pot it' does seem to be the consensus, and in balance it's kind of obviously correct, so thank you all for the correction. And politely too!

Does anyone here ever use the underlying idea of pushing strong hands (say sets) rather than betting smaller for value? I should stress that this is a table at a local room, generally filled with players who are not just loose passive, but actually kinda...dumb. On multiple occasions i've called 2x-3x pot AI on a 2-flush board with as little as second pair and an overcard when closing the action, because the players are so reliably poor that this invariably means flush draw, and they don't generally distinguish between 'flush draw' and 'flush draw with TPTK' for example. Have always ended up ahead.

As a corrolary, would you ever lay down a flush here? I was called in this case, by a small flush, by the one player at the table who would have considered folding it. Is there any FE here against a stronger lineup?

I do have to say that given the stacks (most 50-100), i certainly could have been called by much less than a flush, as was the hope. A pot bet may have accomplished the same thing, but in the heat of the moment i fell prey to the classic "this hand was good preflop, and i'll be damned if i'm not gonna make it good now!"