I am going to offer a different opinion. I believe based on US compyright laws there is a good chance that the gym owns the routine, and therefore has a right to pass it on to another gymnast, and the gymnast can’t take the choreography with her if she goes to another gym.
My thoughts on this are based on dance studios. If a dance studio hires a choreographer, to do a routine for a solo or a group, the students pay for it, but they pay the studio (not the choreographer directly) and then the studio pays the choreographer, so the choreographer is employed by the studio and the choreography is owned by the studio because they were employed by the studio (not the student) to do the routine.
I believe the same would apply in the gym. If the routine was taught in the gym, for a client of the gym the choreographer would be considered to have been employed by the gym to choreograph the routine, therefore the gym would own the choreography. Did you pay the choreographer directly or the gym who then paid the choreographer?