Our range has more equity on this flop. 3better also has the position+initative, so I don't think his ''stronger'' ranges should be an argument for checking flop. This flop is also ugly for our opponent, as he has few strong hands (I personally don't think he flatcalls 22-55, I just don't see how it can be profitabel. He is oop with 50bb and plays vs a bigger 3bet sizing). Furthermore his broadways+scs which are quite a few combos only hit the flushdraw and with something like a7s, he has problems to call because he doesn't know how clean his outs are and the turnshove is always a threat from us.

I added any axs and any pockets to the range below, although I personally don't think he calls that wide.

Board: 5h3h2c
Equity Win Tie
MP2 47.25% 44.48% 2.78% { JJ-22, AQs-A2s, KTs+, QTs+, J9s+, T9s, 98s, 87s, 76s, AQo-AJo, KJo+ }
CO 52.75% 49.97% 2.78% { KK-TT, AdAh, AdAs, AdAc, AhAc, AQs+, A5s-A2s, QTs, JTs, T9s, 98s, 87s, 76s, 65s, AQo+, ATo, KJo }

i agree but i have a few things to add:
-- i think his range may be actually tighter in this spot, given 3b size. I'd say he doesn't have any sets in his range for this size, actually (or rarely).
-- I think we have a pretty strong range advantage, especially if we also 3b hands like A2s-A5s, because now our bluffs have pretty good equity.
-so, I think we can cbet pretty wide here, even with our hands like KxJh

to bet this wide you gotta go smaller. this will lead to a higher flop calling frequency and consequently given the pot size turn is a mandatory shove with whole range, turning the play into a very expensive one.

I don't see how to make money firing 50bbs on this spot, probably against the best player at the table.

he is getting very good odds to peel pocket pairs and I even tough I don't know his game I doubt he is folding 55. cbetting 4.2k seems good.

I dont agree we need to shove our whole range on the turn. There are some hands like kjo and kto that seem like reasonable 3bets, that won't pick up equity and will just check back.
and if you think cbetting our whole range for small sizing and shoving turn a lot is too expensive, then how do you prefer we split our range?
Check AK, AA, some A5s A3s A2s, some KJ KT with no backdoors and cbet TT-KK ,KT KJ with backdoors, some A5s A3s A2s, A4s, ATo ?
(this is assuming we 3b a blocker heavy bluffing range)

I rather play a more simply strategy on those spots when stacks are shallow enough than split my range. Specially on MTTs I rather play whole range aggressive strategies and mix on spots that I am playing deeper.

Chewy flop calling range should have a good portion of 66-88 and he is not getting an amazing price to call turn. He probably is folding a portion of them with some frequency so I wouldn't be too excited on shoving but if you are not bluffing on spots that you are getting called a lot, you are not bluffing enough (:

above, you said "to bet this wide you gotta go smaller. this will lead to a higher flop calling frequency and consequently given the pot size turn is a mandatory shove with whole range, turning the play into a very expensive one.
I don't see how to make money firing 50bbs on this spot, probably against the best player at the table."
so, it seemed u were advocating splitting your range. your now saying you want to cbet nearly your whole range for small size and then fire through with some combos, such at AT, on most turns?

I am not splitting since I am playing a pure strategy with my preflop 3betting range. If I decided to check I am checking my whole 3betting range and if I decided to bet is to bet whole range. To play a bit more simply strategy and maybe making him fold AJ/AQ OTT there is merit to shove ATo and is a hand that still have some equity against 66-88 and possibly some fold equity given image and the turn card.

why play a pure strategy? do u do that always with 3b range?
but i agree jamming ATo on turn here has merit, including with Ah or Th if heart comes in

nope, I tend to play mixed strategies all the time. given stack depth I am not in love splitting my range with a PSB behind OTT. There is basically none bluff that I want to shut down on most turns and also, most of my value hands also doesn't want to shut down on basically any turn. If I had QQ and turn is an ace I might consider checking with some frequency since he may peel two overs + gutter against some of my overpairs (even tough I block a good portion of AQo and if he is calling AJo on this flop he is probably overcalling). A hand as weak as ATo I rather put him in a tough spot and set a turn shove.