We don't know exactly what information or misinformation has been given to the school that needs to be cleared up.
Choralnet might be a good resource for further research for reasons to support why the school should record. For one thing, I've been told that the sound a vocal student hears in one's own head is not the sound that others hear, and that's why so many vocal trainers record as part of their student vocal training.
I remember a lot of wincing among us in high school when we heard what we sounded like.

edit: the school drama teacher might also be a supporter in this because it's one thing to tell a student what they did on stage, and another thing to show them a video recording of mumbled lines, missed cues, improper stage position, looking down, hiding behind other actors....