There is definitely a concern with her teaching herself the skills at home. It’s not just about the skills being safe and functional, it is also about the, being progressive.
The skills need to be done a certain way in order to allow the skill to be build upon. Kids teach themselves to do a back handspring for the sake of doing a back handspring, Coaches will teach the skill in such a way that allows it to develop into and round off back handspring back tuck (layout, twist, double etc). This is why the small technical details are so important. It’s much easier to teach a kid to do a good quality back handspring form scratch than it is to correct one that has been practiced and practiced and practiced so that poor technique has become a habit.
The other problem with self taught skills, is that if kids are doing it incorrectly then certain compensations have to occur to make it work. For example if the back handspring is dome too high and short, the gymnast will need to excessively arch their back to make it work, this puts a lot of strain on the back, the more they do, the more strain and it can develop into stress fractures in the spine.
The coaches attitude seems a little concerning. The statement “I have no control over what you do at home” does seem to indicate that the coach thinks it is a bad idea, but is sort of washing their hands of it. I would have a chat to the coaches and let them know this is happening. Perhaps see if you can organise a private lesson and the coach can actually go through what she is doing at home and see what she is doing technically.
At 7, I do think some well chosen words from the coach, can have a very positive impact. ie Letting her know how much potential she has to be amazing and how she can help herself to become amazing by following the correct progressions to learn skills (but in 7 year old language of course).
Lots of parents and gymnasts choose to ignore this advice, because it doesn’t seem to be immediately hurting them or presenting an immediate problem. But I have had the unfortunate experience of seeing the end results of poor technique practiced for years, and it very rarely ends well.