Sammy, I'm on your side with fielding pitcher question. When most of these pitchers are young, they are often some of the best, most practiced players on the field. (not always of course) When they get to college, suddenly they are pitchers only and it's assumed they can't field, at least as well as the regulars. (When was the last time you saw a MLB pitcher catch a popup?) While they may not take as many balls as the regular fielders in practice, they can still play and should be used any time it's 51-49 or more the pitcher's ball.
Case in point, earlier this year, there was a popup in the infield to the right and behind the circle. SS was playing deep and called for the ball, sprinting to get it. As you can guess, she fumbled the running catch while the pitcher could have had it with normal effort. (not sure what happened to the right side of the infield.) Deffered to the SS because of the "no field" concept. Similar popup a couple games later, this time in front of the pitcher. 1B running in and about to call for it, when we all hear MINE MINE MINE quite loudly from the pitcher. Easy catch for the out and lesson learned. Didn't see the coaches reaction, but the first baseman was looking at her like she was a martian!
Probably a case of catch it-- nothing said, drop it --- woe unto you. Good for her tho, fool me once shame on you, fool me twice, shame on me.
I guess my final answer to the original question would be to train your players to think on their feet and make the play accordingly. Give them the physical training to have the tools to do what is needed in that play, let them decide in the moment which method to use. ie Practice all three ways to make that play, and that gives them the best chance to make it when it counts using whichever method the situation calls for. Hard and fast rules sometimes lead to a botched play.