Here are a couple quick assessments and tricks I’ve learned in my yoga teacher training school that might also help. Since it’s yoga-based, it’s relevant for push-ups done on the ground, but not with Equalizers, Perfect Push-up, or other equipment.
1) Look where the wrist crease is. Ensure the exerciser’s wrist creases are pointing forward. Then place the elbows and shoulder blades after positioning the wrists.
2) Instead of right over the wrists, place the shoulders very slightly behind the wrists. Not a lot, just enough to soften the angle of pressure.
3) Make a little pad out of the yoga mat and place the heels of the hands on the raised pad. Again, softens the angle of pressure.
4) Ensure client is pushing down with entire hand. No white-knuckled finger pads. Not just in the heel of the hand. Pres the top of the palm down and have that take some of the weight.
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.
I would add that the problem is most likely that the wrists/forearms need to be addressed first. There could be a number of reasons for the pain, but you can’t find out why without doing some wrist/forearm assessment and strengthening. First they should be cleared to work on their wrists by their doctor. And recommendations should be sought for proceeding.