Honestly making it socially unacceptable is probably the only solution. Not sure how that happens though...
Trying to police the shape of the camera is silly.
The 1st Amendment does not apply to private property. Property owners, even owners of public utilities, can absolutely set boundaries and trespass people. In some states, property owners can literally shoot trespassers to death.
In the case of truly public spaces, historically it's considered that other then in very specific situations (bathrooms for example) you have no expectations of privacy and can be recorded regardless of the shape of the camera.
On private spaces, which people constantly mix up with public spaces, you're absolutely correct that it is able to be restricted and can define very granularly what's acceptable. A plane is a private space. They can restrict anything they want (limited by protected class/characteristics in the US. Other countries may vary).
It also doesn't protect from any consequences of your free speech of course.
I'm not a legal expert but the first amendment doesn't generally give the right to secretly record people, especially in places where there are already laws about hidden cameras, recording, privacy, etc.
It's not about the shape as much as the intention and execution.
https://en.wikipedia.org/wiki/Reasonable_expectation_of_priv...
In which case, the First Amendment still probably wouldn't apply, unless we're talking about laws against specific musical content, since content-neutral restrictictions on public speech are broadly legal.
Why?
She purchased some stylish ones from Asia, so it almost looks like part of the uniform. At least the manager is fine with that.