Let me play devil's advocate for a moment: I can see why a gym would have such a policy. If there are very specific progressions you have planned, and especially if you are a more conservative coach, it's reasonable not to want your kid going to a less-conservative gym where they might just let the kid chuck skills you'd rather have them wait to do.
Generally, the gymnast's own gym will train with the goal of sustainable long-term development, whereas from a PR perspective any camp's goal is for them to go home with as many new skills as possible.
For example, I prefer not to teach a tucked flyaway off bars until the gymnast is learning double-backs, because teaching a layout first emphasizes a stretched position on release and avoids any early tucking action. I am very very picky about the tap swing going into the flyaway, and often will not allow kids to work the flyaway at all until the tap swing is extremely strong, even when the gymnast is physically capable of releasing and flipping to her feet. If I had a gymnast who was part-way through the progression who then went to a camp where they did not follow this model, she'd very likely learn her tucked flyaway easily and quickly there -- but this would in some instances negate the benefits of learning the layout first.
So I can see why a gym would have such a policy.
That said, I think outright prohibiting a gymnast from going to a camp at another gym is excessive.