Isnt your son relatively new at pitching?
This is the typical scenerio of a coach using a pitcher poorly. If a ML starting pitcher gets pulled early, does he go out and try again the next day? NO!!!
His shoulder hurts because he hasnt developed the muscles that help support a pitcher when they throw, that's not only in the shoulder, but the core muscles and the butt and legs as well. Until that develops, he will be most likely throwing more using his "arm" than his entire body. So that means, "take it easy".
Playing first means he hit as well, that was added stress on his shoulder. At 12/13, he should have some type of recovery program and the pitching instructor should help with that. If he wasnt hitting his spots, then more than likely he was having issues with his mechanics, maybe you need another insturctor?
Even at that age, if son pitched he was not in the game. And he never pitched the next day, regardless.
As far as to ice or not to ice, that is a personal decision, son always iced, but he also threw much harder than most.
Yes, he is new to pitching, you make a totally valid point that he probably doesn't have those muscles developed and likely threw with a lot of arm.
I asked him why he thought he missed on Saturday night and in short he said because I didn't think the catcher could take full force, but trying to throw softer just screwed everything up. I told him that coach would probably have him pitch the next day and he needed to pitch with his natural movement and if the catcher couldn't handle it then that was the coaches problem NOT HIS as long as he was throwing strikes or near enough. Sunday you could see he just let loose, and didn't walk anyone in the 4 innings he pitched.
I think the muscles not being there is dead on, I'll look into getting some of those bands. Thank you.