Farewelltoarms isn't your interpretation of the comments about primary level schooling a bit reductive? I don't think it's as simple as suggesting kids in primary schools are all running riot just because some suggest private schools are "better".
Speaking from personal experience, I sent both my 2 to the local primary, a nice OFSTED "outstanding" school. It was ok. DD who is "gifted and talented" (level 5s at end of year 4) has been bored in school for ages because the school's idea of "extension work" for her has been just to let her do the work of the year 5s and 6s. In the last year, when doing that work itself hasn't been a challenge, the approach has just been "what more can we do?"
On the other hand, DS is a much more middling child, can do well but not like his sister. Has finished year 6 with a perfectly respectable level 5s and a 4. But he has had no homework at all for the whole of year 6, has not been encouraged in school to stretch himself that bit further to get the 4 up to a 5, and from what I can see, has just loafed around in class while all of the effort of the teachers has been focused heavily on the few kids at the bottom of the class who were not going to hit level 4s across the board. All the attention is on dragging those ones up to the minimum "good" standard so the school can look better on it's SATS results.
I understand this is even more so at secondary level with the holy grail of 5 A-C at GCSE. If my DS - who given the chance is quite happy to coast along and do "ok" rather than go the extra mile to do his "best" - were to go to our local (also "outstanding") comp., he would probably leave with some A-C grades but I know he can do more than that and that's why I'm sending him to a selective private school instead. DD is also moving and can't wait "to do some interesting work". I will have less than £300 left at the end of the month like the OP but just consider the smaller class sizes and higher expectations in the schools they are going to to be vital.
Little story about the expectation levels: when DS was approaching the time for the exam for the school he's going to, I got some Bond papers for him to practise. He was routinely getting 60% on the maths (min. 70% needed). His primary school had always given him fantastic reports for maths, so I wrote to his teacher with a copy of a paper he'd done asking if this was the level she expected of him as a "good" maths pupil, as it was nowhere near the required pass mark. I got a letter back which simply said "the expectation levels are very different". She was happy to leave that as her answer! This is in an "outstanding" school, remember.
I'm sure it all evens out in the end, cream rises to the top in any school, etc. But I want my kids to really enjoy school and find it interesting while they're there, not just coast along while the school concentrates its efforts on the kids at the bottom of the class. That's why I have decided to move them to selective private schools and I think many people move for similar reasons.