I like a lot of these previous posts- modifying the movement, myofascial release, wraps, etc. are sometimes helpful in preventing wrist pain and if it persists the client should consult a health professional.
One thing I will add that helped me with several clients is to make sure the client’s palm is in the proper position. Their weight should be distributed evenly through all “4 corners” of the palm. In my experience women (and some men) often lift the portion of their palm directly below their pointer finger and/or at the base of the thumb. This increases strain on their wrist joint as well as their forearms.
Another pushup modification I’ve used (if wrist strain is the only issue) is having the client do the pushup on a Bosu (on the floor) or a stability ball (against the wall). With either piece of equipment they can turn their fingertips outward and reduce wrist strain. It’s also a softer surface on which to rest their hands, which sometimes helps. They can still modify a full-body pushup to their knees or put the ball/Bosu higher on the wall to reduce the percentage of their body weight being used for the exercise.
It depends on the person and their injury. A physician would always be a good idea to start with, but other options would be to either eliminate pushups for the moment and see if the pain goes away or start using other equipment (like the ones you have mentioned in your question) so it can take some pressure off their wrists. Elevated push-ups could be another option for your client or even the usage of bands and/or TRX to add some chest presses. Once they get stronger doing push-ups on an elevated surface (bench, steps, boxes, etc.) then you can start moving them lower to the ground until they can do them without any discomfort and/or pain.
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.
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.