Some academies are ridiculous. Well done OP for moving your DD!
My DC have loads of friends who attend top private schools in London because they do orchestras with them. In many of these top private schools it seems the kids are allowed Air Force trainers as school shoes, short skirts, not a big deal if something is forgotten either and they can access their phones too at certain points during the day. Yet these kids still do incredibly well academically, sports, music, drama, the lot because they are treated like individuals and valued.
In our grammar schools there are no such very petty rules either. They do set a lot of homework and high expectations, but the rules are really not as petty. Mine have never had detentions. Instead I might get an email from a teacher to me stating your child has forgotten their physics book 3 times (turns out it is the same colour as the Chemistry book so he got confused). The emphasis is on working WITH the parents, not AGAINST them. I think the same must be the case in private schools, the parents are valued clients, the grammars know we are educationally supportive parents.
Children with ADHD need clear routines and checklists at home/in their school bag and pack the bag the night before, go through all the lessons, lay out the uniform. They need to develop routines and strategies in a supportive manner to deal with their ADHD as best as they can. If they are not supported, then they go into fight and flights and everything goes wrong.