instead of covering gymnasts head to toe to combat "sexualization", how about we get rid of the mindset of skin inherently being sexual. private areas being covered is obvious, arms and legs being covered doesn't make sense because they aren't private parts, and who wants to be performing...
for anyone dealing with body image and/or are having their period, I would recommend low-cut black dance briefs and/or semi-translucent/sheer skin tone tights (sheer tights are more breathable)
I think you should find a new gym, completely unacceptable behavior from the owner. Regarding the gym not allowing parents, I think that is pretty normal, as it can be a safety hazard for parents to be on the gym floor. If you want to spectate, you can ask if you are allowed to safely spectate...
Shorts or pants can and should be used for warm-up since they help you, well, warm up your muscles, but after warm-up, shorts should be removed to allow for maximum degree of motion, less weight and less air drag, also helps to keep the gymnast from overheating (in extreme cases), this is also...
the lululemon boogie short could be a good choice, you could also try low-cut dance briefs (which are very similar to shorts) as they also offer more modesty than a conventional high-cut leotard
EDIT: If a gymnast does choose to wear shorts over their leotard (which is okay), I recommend wearing form-fitting short shorts, as they are more modest than a leotard but allow more movement than regular length shorts, so its sort of gets the best of both worlds
Its a lot harder to use the restroom in a unitard or long tights/leggings as you have to fully remove it, especially during competition where you always have to be ready, a legless leotard is much easier to use the restroom in because you can just move it to the side. Shorts over a leotard...