Just to echo Dahlialover: I was that sort of gymnast and while there were lots of things going on in my life that later led to mental health issues (eating disorder, but not general anxiety), I am sure my fearfulness at gymnastics had absolutely nothing to do with that. I also was afraid of vault (only vaulted without spot at meets), took forever each session to dare put my feet on the beam on a cartwheel (could do hundreds without falling on low beam, but always bailed out on high beam), and flat out refused to do back walkovers and back handsprings on high beam (the end of my gymnastics "progression"). I was always afraid in a specific way - cartwheel on beam was splitting the beam, most backwards things was breaking my neck, etc. - on the other hand, I was never afraid of something like a front somi because if I fell, I would land on my bum with no harm done. That still seems sane to me.
I now, at age 47, climb and everybody laughs at me for only top-roping and refusing to lead or to climb outdoors, but to me it seems like a no-brainer to avoid any additional risk I can. In life, I am wary of flying because, while it is super unlikely, a plane can crash and that would be fatal, I am afraid of cancer, and one ingredient of me not driving (outside lack of necessity where I live and environmental concerns) is the worry of killing someone, but I have never understood fear of (non-poisonous) spiders which have no way of harming me, and while I may be nervous before giving a big conference talk (scientist), I am not anxious/scared (worst case, I make an idiot of myself but I'll survive!) - you get the idea.
If your child can get interested in other sports, that would be great, but for me, gymnastics was my only sporting love as a child, despite my lack of progress due to extreme caution. It was only as an adult that I started to enjoy swimming, dance, climbing, etc. So she might insist on gymnastics and just progress very slowly/stall and that's okay. What helped me a lot was a change to some low-level optionals (maybe a bit like excel, but this was in the late 80s/early 90s and I am not from the US, so don't know the system well), where I could expand skills laterally and find additional non-scary skills.