Doing pushups and burpees in a kickboxing class sounds really tough. If I have not done burpees for a while they give me DOMs in many places including my traps. Neither pushups nor pullups have ever managed that.
A few years ago I gave up most weightlifting and now do mostly bodyweight stuff. The only lifts I still do are trapbar deadlifts and shoulder presses. I've lost size in my chest but all my other muscles seem just as big.
I love bodyweight exercises over weights for sure.... although I guess the only real bodyweight exercise for the shoulders would be handstand pushups? So shoulder presses do seem like a better alternative.
If I can get my elbows under control, I really would like to do obstacle course events (like the Ninja Warrior tv shows) in which case I want to be strong but not heavy... which bodyweight exercises are really good for.
Yes, I prefer shoulder presses over handstand pushups. And regular bodyweight squats are kinda easy so pistol squats are usually recommended as a more challenging alternative. But I prefer to do trapbar deadlifts rather than progress to pistol squats.
Good Luck on rehabbing to be able to do obstacle courses again. I have watched Ninja Warrior on tv and enjoyed it. But I don't think I am acrobatic enough to do a lot of the things I would need to do in order to make it through a course.