It currently does absolutely nothing and because of that it’s very unintuitive. Let me give you an example: You’re used to enemies staggering when you push them; Enter ratman Bob who is at the last frame of his climb animation, which renders him immune to push effects. Pushing him does not what you expect it to do (send him flying back) and instead he attacks you dealing damage right after. This is a case of bad user experience, because the interaction does not work as you’d expect.
Also showing doing something does not that it trivializes ledge fighting (eg. in the case it does very little and attacking is still more useful 100% of the time). The op never said how strong shove vs climbing enemies should be or how it should work, just that it should do something and gave ideas what it could do.
Adding options to players: eg. flee, shove or fight adds depth to the game play because the player has more ways to approach each situation making it harder to determine which is the optimal one. Optimally there would be no “go to” solution and each option had its best moments.