Sorry, I got kind of behind on reading this thread.
Briefly: nothing is wrong with my wrists that I am aware of, in terms of medical issues or strength The closest bone I have broken is in my ankle.
I find if I am riding flat bars on anyone's mountain bike (mine, my adult kids, recent demos from Santa Cruz, etc), I have the same issue, so I think it is roughly bike-independent.
Right now I have a 2008 Trek Fuel EX 7 that mostly sits in the garage, unused. It has a relatively long stem and short bars, and I put bar-ends on them, internal to the brake levers, which offers a bit of relief.
I have a steel drop-bar steel mountain bike with front shocks and (alternatively) a rigid fork. I pilfered the drop bars from it for my touring bike, and at the moment it has Jones bars on it, but I think the geometry is wrong (Jones Bars assume a reach similar to flat-bar mountain bikes, whereas my drop bar mountain bike has a shorter reach). I've also tried Jones bars on the other bike, and while it is definitely an improvement, in retrospect I probably should have gone for something like the Whiskey Winston bars