I do despair of threads like this. People are so wilfully it seems ignorant of the reality of those in differing situations to themselves.
"Contact some schools that offer boarding"
The OP says she has four children, and 90% of the country cannot afford bog standard fees, let alone boarding. I'm sure she's well aware these schools exist, and has dismissed the possibility because she knows her own budget.
"Some schools are so precious about space"
No understanding whatsoever of the reality for inner city schools. Our school was built to admit 100 pupils per year, and we're up to 150, on a tiny site. We literally cannot fit any more pupils into classrooms. We had one way systems even before covid, because it's dangerous to pass on stairs with the volume of pupils.
"His current school have to keep him"
No they don't, they have specific published entrance criteria in the event of more applicants than places and need to stick to those or (rightly) those not getting a place can appeal.
My school could fill the 6th form three times over with those that meet the criteria. When places have gone, they're gone. You can't just ring the head and explain the situation. We can't just accept an extra pupil and split the class- we've no spare classrooms, and no spare teachers.
We had parents yelling and remonstrating with the HT on the street outside school last week at enrolment when staff went out and told the queue of waiting children and parents that we were full.
"Most people would be so pleased to have those results"
Yes, of course they're good results...but if a child is capable of 10x 9 at GCSE, and gets a mixture of 7s and 6s, then no, they're not good. Conversely if a child was expected to get 5x 3s and managed a mix of 4s and 5s, they'd have done brilliantly and as their parent I would be so proud of them for exceeding expectations. Context is everything. For all we know this child, while capable of 10x 8 grades may have lost his father six months before his exams, be a young carer, have a sibling with autism whose been on "home learning" for the past 15 months, etc etc etc.
But, I know this is AIBU, so par for the course. 