Not even being willing to have her interact with the trainer without you physically right.there. is coddling.
Is she home schooled? A lot of what you’re describing as a no-go is happening every day in the traditional school setting. Interacting with kids and adults, without mommy/daddy right there.
I hated organized sports as a kid, but that’s because I wanted a horse. (EDIT: Guess what? My parents still made me participate, because part of life is doing things you don’t want to do, that are good for you in the long run…) Be careful that your own mental state is not impacting what she “can and can not do”. It’s fluid, and she has to edge up to the comfort zone line to expand it.