she needs the executive function and independence. Not relying on parents as much as she has been. Taking accountability.
This along with other things is really alarming me.
I know we don't know the whole story OP and you are wanting specific factual information about only the bit that matters to you right now, and us asking about other elements of your home life appears to be frustrating to you as it seems both emotionally challenging you and not relevant. (This is exactly how the autistic people in my life would behave when asking this question by the way, just leaving that there)
It feels ethically not right to me to give advice about boarding schools for a vulnerable neurodivergent girl without understanding the context and what other pastoral care she has now. I can see you're not wanting to share this, or maybe you don't think it relevant to us.
But you are describing things as if they are "things she needs to learn" - and as if she just needs to knuckle down. This includes manners, executive functioning, interest in learning. There has been no explicit recognition from you on this thread that these might be things she needs additional support to learn. From this gap, I get the impression that her needs are not being met, because if you're the sort of parent who understands and meets those needs, you say so. It would have been the first thing you wrote in your OP.
Even a small note from you, demonstrating that you understand autism, that you are parenting her in a neuro affirming way, rather than wanting her to comply with neurotypical norms, would give commentators on this thread some comfort that their advice will not lead to harming this child.
At the moment I have a strong sense of moral injury. This isn't reddit, I feel on Mumsnet that words have weight and we are not just shouting into the void, that real people could be helped or harmed by whatI write. So if I say to you yes Shrewsbury is a great school - or even if I name other boarding schools which are genuinely great for autism - I might be instrumental in you taking advice which would harm your daughter.