Catchers put in just as much work, if not more than pitchers. For starters most pitchers can’t hit the broad side of a barn so the catcher has to chase the ball everyplace. They do all of their individual work AND catch for the pitcher at her lessons. They arrive at practice early and leave late to lug all the heavy equipment around. Catchers drag that huge bag around that wheels fell off in the first week while pitchers have their parents carry their bag so their arm doesn’t get tired. The closest a pitcher comes to running is sprinting out of practice early to get out of helping put things away. And the biggest difference is catchers parents don’t make sure everybody knows how much harder their kid is working.
Yes good catchers do all those things you listed, my point was I see a lot of catchers do not. No player on our team is allowed to have their parents carry their bag, ever.