If one child is told she can't compete, that possibly is the end of the world. Particularly if she is also going through the hell of being othered, hated, ridiculed, bullied, picked on etc. That sport might be all she's got.
Kids are told they can't compete all the time (as are many adults). Like many adults, there were plenty of things I couldn't compete or take part in as a child for reasons entirely or mostly out of my control - I moved after the season started or was going to move soon, I didn't make the team, far more girls than boys tried out so more of the girls didn't get a part, I didn't have the grades to be allowed to take part, and so on. If a child struggles to cope with that, the answer is therapy and an environment to better develop coping strategies beyond just one sport, not changing the rules thinking that somehow will automatically improve someone's wellbeing. It really doesn't work that way - sports can do a lot of things, but for kids going through hell, it's not the fix being described. That hell just changes, even taking over the initial high of being able to take part, without a lot of support that sadly not enough people are getting. Just bending the rules is a cheap, heartless attempt and pretending to care, in my opinion, it doesn't really do anything for any of the issues described.
There was a time wrestling was pretty much the only positive thing for my life (which, with the general culture in wrestling being brutal and my team nickname being Miss Worthless which was probably one of the nicer names going around, probably says something), but if I hadn't been allowed to compete because of my sex [and there were times I was not allowed to compete due to my academic standing], the answer was for the adults in my life to help me cope with that, not pretend my horrible circumstances which did involve bullying, neglect, othered, being violently hated, and a lot more meant the rules shouldn't apply to me. This was difficult because I was a messed up kid with a lot going against me and I'm not sure I'll ever entirely heal from it, but that doesn't mean I deserve to be part of it without meeting the rules. My pain and circumstances give me a particular perspective, but it doesn't change that competitions aren't a right. Being respected, living publically without fear, yes, but not sports competitions.
Mixed-sex competition and the training for them starting from around the age that child has more risks than the single-sex competition. It's well known that it's hard to get the best out of girls training for mixed-sex competition because of those risks - we have to push harder to keep up which means injuries are more likely. It seems far more unfair to let the child to compete as a girl now and have that taken away later. While I'm not against mixed-sex competition as an option, single-sex sex should be the standard, especially going into and through puberty, and it would be unfair to the girls to make it mixed sex solely based on how a male child looks. I knew plenty of waif feminine looking wrestlers in my day and see plenty of them now that could still throw anyone in their weight class across a room and did so merrily, something even at my peak I was never able to do safely due to my female muscle-to-fat ratio and skeleton.
And, as a mixed-race American, the misuse of Rosa Parks is horrible in appropriating the black and American civil rights experience - there is enough shit in this country that is actually relevant (and, no, it was not everyone, maybe learn more than the snapshots of civil rights history. Rosa Parks was not the first to not move and she only had an impact because she was a long-term activist with a whole community of people able to support her and the bus boycotts that came after with replacement services and care) . Women have fought for ages to have women's sports. Look at the history of women's exclusion from sports in this country. Look how long the FA banned women's teams from their fields and stadiums - it's well within living memory. Look at the fight women are still going through for the same standing in sports in the UK. It's absolutely disgraceful to use something White people in power put on others as any sort of comparison to something women are still fighting to have around the world.