Do you think thats the most profitable way?
All-in with bigger pairs preflop is one way to play, but I don't usually like this. Preference, but I seldom like being All-in; even with AA. Obviously, it depends on factors like effective stacks though. If we are say 20 Big Blinds deep, then yes, I'm usually all right with getting my entire stack in before showdown. However, being 100+ Big Blinds in (deep stacked for sure), then just shoving (even AA) probably isn't the most efficient way to play. This just makes it too easy for everyone to fold and give you the cheap pot. I'm not saying to slowplay the Aces, but just generic raising and betting for value should usually be sufficient; I'm not always looking to go All-in.
Don't slow play your AA's.
Occasionally, I will slowplay AA. This is more of a possibility with:
-stacks super short because you can get all of your chips into the middle anyway
or
-an aggressive opponent or table that might make trapping justifiable
but yes, generally, we want to be playing AA straight-forward and value betting/raising and re-raising to isolate and/or build a pot.
As for my analysis of this hand, I think you played it right for the most part. Just unlucky that the opponent Flopped two pair (and Doyle's hand of T2 no less
here).
One small observation is that you made it 3x when you raised, but there was one limper before you. If you generally raise 3x, then usually we can add 1x per limper. So a 4x raise might have been slightly better to discourage the limper from continuing. I won't critique every street of this hand in detail (others have done that enough already), but I think it is just an unlucky case where AA lost and we should move on. AA does get cracked against a random holding about 15% of the time you know. It isn't a sure fire thing and the opponent flopping two-pair is one way we could realistically lose with AA, but if the opponent is calling raises with T2 frequently, then we will probably pick up chips on them long-term.