your argument is flawed:

if the situation was repeated indefinitely, you won't have Ax every time.

WHEN YOU HAPPEN TO HAVE Ax, HE WILL LOSE. BUT HE LOSES LESS BY CALLING THE PUSH THAN FOLDING

WHEN YOU DON'T HAVE Ax, HE WILL WIN


If he somehow knew you had an ace, then his play is wrong. But he doesn't. As long as his gain from the times you don't have an ace outweighs his loss from the times you do, his bet is +EV.

read greenstein's Ace on the River, specifically his comments re: distinguishing the 'correct' play from the 'perfect' play.

As for your final sentence: what are you going to do the times you call without an ace? Are you going to bet then? And if you bet then, but check raise Ax, they ain't folding.