So I think the situation that Gilly describes, and another poster upthread who talked about a specialist ASD school, are somewhat different - a genuinely specialist school offering an education that simply isn't available elsewhere. I still think that in your shoes dh and I would relocate to live close enough to the school so they could be predominantly day pupils, but I can see that a genuinely specialist education is a strong pull factor. I don't think that is what most posters are concerned about.
However most boarding schools aren't like that - yes they have good education and "great facilities" but something similar could be found in private day schools or grammar schools. I simply can't understand the choice to send away to boarding school rather than a good day school.
We would prioritise moving and living with our children over living in a certain area ourselves. If we lived in an area with only crap schools we would move, change jobs if nrcessary, have a smaller house in a better area, rent. If really necessary dh would work away and I would live with the children near the school. Dh and I would be much more able to cope with separation than a child!
Even if my young child said they wanted to go to boarding school, I would take that with a pinch of salt. Young children aren't always able to fully appreciate the ramifications of decisions and don't know how they (and other children) will change as they get older. As a child I was desperate to go to boarding school - I loved boarding school books, I loved primary school and my family life wasn't that happy. However I'm really glad my mum couldn't afford it, as actually it wouldn't have been good for me. Private space, a world outside the intense world of school friendships, down time with family where you can be yourself - really important (even though my home life was stressful too). I now look back and realise I was anxious and uneasy a lot of the time at school, and struggled to fit in. I wouldn't have recognised this at the time, and I certainly didn't tell my parents, I always said - and mostly believed - that I liked school and had lots of friends. I just don't think most young children have the emotional maturity to make these assessments.
I'm not talking about your daughter here Gilly as yes she does sound exceptionally driven and the school is offering her something another school couldn't. So whilst I wouldn't make that choice myself I can understand your reasoning. I don't think your case is representative of most children at boarding school.
Of those I know who went away to boarding school, some say they missed their parents horribly but put a brave face on, and they would say they thought it was the wrong decision. Others resolutely insist that it was Fantastic and they would definitely send their own children to boarding school, nothing else would do. When you ask why it was so great, the reasons are odd. Firstly the facilities, which can be accessed as a day pupil, so not a good reason. Secondly they often talk about being made to stand on their own two feet and be independent at a young age. I don't think forcing a young child to be independent at an arbitrary age is desirable. Of course their peers who went to day school have also managed to grow into independent adults. Why force it early? In fact many of them have gone into the military, which is a bit like boarding school, so not fully independent....
One said that he thought boarding school was character building as he had to be a "fag" (errand boy) for a senior pupil - clean his study, do his errands. This scheme was compulsory. In exchange the senior boy gave him some protection from bullies. As an intelligent man in his 30s he thought this scheme was excellent and he was pissed off it had been stopped before he became a senior. He just didn't see it was horrible, to have an 18 year old bossing around a new young boy, with the hope that if he cleaned well enough and earnt favour, he might get some let up from the bullies.
Of those I know who went to boarding school, they are "successful" - good careers, well off, wide circle of mates, attractive spouses - but lacking in self awareness and empathy, denying any weaknesses, and determined to Cope with a grin and a joke in any circumstances. These were ex's and previous good friends so I knew them well.
The exception was a girl who was in boarding school as SS had taken her into care, her grandparents had her in the holidays but couldn't year round, she was delighted as it was better than foster care in her experience.