Similar to how chain clothing stores want their employees to wear only that chain's clothing purchased with the emoloyees' wages, sounds like the owners want their employees to spend their own money at that gym.
I don't think this is it tbh, because based on what was said it seems like they would've thought it was fine if she was doing regular activities at another gym since they were interested if she did yoga, barre, Crossfit, and other stuff elsewhere, and were obviously disappointed that she did not.
Either way, the decision doesn't seem linked to OP's personal fitness abilities and really should've been taken care of through the hiring process, as you note.