This is consistent with my experience as well.
My post became longer than I expected to write
TL;DR: Similar problems, fixed diet, lost a bunch of weight, started exercising daily after achieving normal weight, all problems thing of the past. True for > 5 years now.
Both of my wrists were broken in my youth, in separate incidents. To make matters worse, one didn't heal correctly and needed to be re-broken.
Until my late 20s I obsessively spent long sleepless days at the computer, all classic qwerty keyboards, either model M or classic thinkpads later. My wrists would bother me at the computer, but they also would hurt when it rained, or other random moments. When I would try do some push-ups, my wrists would hurt like hell before I got anywhere near the expected muscle fatigue. This was generally accepted as expected and how things would be forever (or worse) thanks to the broken bones, according to those I consulted (doctors and family).
Except I wasn't minding nutrition, physical activity, or sleep at all, and lived in an environment (fly-over state) with an unhealthy food culture and awful climate. Without making a conscious deliberate effort, the default outcome was to slowly become fat and sick.
Fortunately a startup relocated me to the bay area, where I discovered even walking up a mild hill had become a strenuous, exhausting activity. Having a great climate, accessible local produce that actually tasted good, and a newfound interest in enjoying a life beyond the keyboard with the abundant outdoor beauty and activities, improving these facets of life came easily and I dare say I played little role beyond cooperating.
For ~5 years now I've been doing 300-600 push-ups daily, two sets in the morning and two in the evening. My wrists never bother me, not during push-ups, not at the computer, my injuries are a forgotten memory that I'm only reminded of in threads like these. There's usually also some sets of ~12 hand-stand push-ups, and holding a hand-stand for a minute while doing leg-spreads against a wall.
The only reason I'm able to sustain this daily routine is my weight got down to ~165 (5'8") from ~225. This didn't result from exercise, just vastly improved nutrition fixed my weight. After my weight was normal, the activity started because it was easy, fun, and painless. Words can't describe the pleasure of doing calisthenics with a body 25% lighter than it used to be. My experience exercising now is a daily celebration of being alive and well, and the feedback these activities provide keep my life in check.
I've also noticed that as I became stronger and more adept at push-ups, I was less abusive to my wrist joints and instead distributed the weight across my fingers and even slightly elevated the heel of my palm rather than leaving the area inactive (letting the wrist joint just take up all the abuse as I repeatedly drove my weight into the floor through it.) This wasn't possible for a long time because all the other muscles had to develop and catch up to the chest which was always capable of doing 20-30.
For those of you saying your wrists are ruined from daily push-ups, I'm dubious of your claims. My wrists seem to only become more resilient as I do more push-ups. If you're overweight, have gout, or something else caused by poor nutrition then you should probably address that instead of ceasing the activities altogether. There are more or less abusive techniques to push-ups, and if you're doing them daily in large quantities, you should be able to develop a painless technique, be nicer to your joints by activating the muscles in the area.
My $.02.