If you are a serious baseball player (heading for college or Pro), you are a pitcher or you are not! Not the pitcher today, then tomorrow's shortstop or catcher. Pitchers pitch today, then rest/ slowly prepare for their next game in a few days. You can't be a race horse today and a work horse tomorrow!!
Yes, the throw is different; different mechanics. Pitchers may throw 100 pitches today; most in earnest. Shortstops and catchers will make throws in earnest too.... but maybe about 12!! So, the pitcher is at a higher risk for injury.
If you believe your future is pitching, then train that way. If not, focus on the position(s) that provides your best future in the game. Any coach/ manager that does not agree with this, is not acting in the best interest of the player. I believe a discussion with the coach prior to the season is the best way to determine the path. If there is a disagreement, it may be a bad match...move on. Why ruin a promising shortstop or catcher because the current team is short on pitching?