A sore arm is like a headache or a toothache. It can make you feel bad, but if you just forget about it and do what you have to do, it will go away. If you really like to pitch and you want to pitch, that's what you'll do.
Twenty games is the magic figure for pitchers - .300 is the magic figures for batters. It pays off in salary and reputation. And those are the two things that keep a ballplayer in business.