Just curious, but where is this written other than your post? I have never read it anywhere. Not in any rulebook nor in any laws. Please let us all know where this is stated.
I purposely used "similar" and not the "same" concept to make a point (of course there is not a legal issue here, but more ethical). If someone is paying for lessons, I would hope that the pitching coach would be looking out for the student and not undermining her behind her back. There is a lot of personal information that gets shared with a pitching coach, including whether and how much time she is putting in to get better, etc. and I would not want that information used against the student by the team coach. The student needs to be 100% comfortable to share information with her PC and if that trust is broken, the relationship is compromised.