Thank U 4 Posting.
The result answered your question. You were all-in with a chance to triple up and you still made the money. How could folding be +ev compared to that outcome?
Our number one goal is winning the tournament.
These questions about AA are not where we should be focusing our time. What you need to be working on is how do I not get down to 3 BB. You need to build your stack or bust out way sooner than this. It does happen that we were trying to build our stack and ended up in this spot. When that happens the AA is a no thinking necessary call.
Cashing only becomes relevant when it is a 50-50 decision on how to play a hand on the bubble. AA is never a 50-50 decision. AK AQ QQ JJ 1010 could be but not AA, KK could be but not often?
So to be clear 50-50 decisions happen when you have 50% equity or < and the pot is 1-1. Not when you have 50% equity or > and the pot is paying 2-1 or >
That you lost does not matter. We do not play to win hands we play to make the best choices. The math says the best choice here is to call every time. So that is what we do.
Hope this helps