I'm not sure what the point of this thread is?? It sounds to me like that coach was trying anything to get into the heads of your team. I'll give him the benefit of a doubt and assume he just didn't know the rules and not that he was being a total jerk in a 12U game.
Your suggestion that only a F2 or F3 can use a mitt is correct. However, I'm not sure that had your son gone to the mound and begun throwing warm-ups, that the choice of his glove would have mattered. I think at that point he would have been assumed to be the pitcher for the upcoming inning and should have been required to use a glove and not a mitt. However, the fact that he walked across the mound and threw the ball to the pitcher coming out of the dugout does not show any intent to be a pitcher.
Here is the rule-
(a) If no announcement of a substitution is made, the substitute shall be considered as having entered the game when—
(1) If a pitcher, he takes his place on the pitcher’s plate;
I would have used this as a teachable moment- only the F1 steps on the mound unless there is a meeting. When coming on and off the field you go around the mound!