This would be REALLY expensive, but would solve it:
Have robotic cameras all around the fields. The cameras would be controlled by FIRST, for FIRST. No camera crews within 50 feet of the field. If they want better shots, then they buy the footage off of FIRST's robotic cameras. And the robotic cameras don't have to bee that big, maybe the size of a coffee mug or something.
(And I agree completely with your post
)