If we mainly play to min-cash, then we should play DoNs or perhaps 9-man SnGs. If we have decided to enter an MTT, we should focus on maximizing EV and not on min-cashing. This does not mean, that we should go totally bonkers and only care about winning the tournament. But if our
equity is high enough, which it always will be with AA preflop, that overrides the fact, that chips won are less valuable than chips lost.
The only reasonable exception might be, if we somehow won an entry to a high stakes tournament, that we are never going to play again, and where even a min-cash would significantly improve our financial situation. If we are behind on our mortgage and risk losing our house, and a min-cash could have solved that, it would be reckless to not secure it before trying to accumulate chips to run deep in the tournament.
Another example is, what David Fishman did in the TV-show "big game" after being up more than 100.000$. The rules did not allow him to leave ahead of time, but now playing with his own money, he knew, he would not be able to take any risks, and the pros would likely understand this and relentlessly push him around. So he wisely decided to just fold every hand and pay the blinds and antes, until the game was over. He famously folded AA, and the flop then gave another player quads.