I agree that the ability to push is a foundational movement that we need in some form to do our activities of daily living.
(a) not everyone needs to be able to push up their full body weight for daily living
(b) until an individual knows the approximate cause of the pain, either in the wrist itself or if caused by movement / position in the joint(s) above, it’s not wise to just keep trying it. The client might legitimately be helped with strength or mobility at the wrist, but if it’s a bony limit they’re hitting, that’s only going to be aggravated by successive attempts.
(c) there is more than one way to skin a cat, so you could come up with other ways to build this functionality without a push-up.