State schools are slow to give places, so parents panic. Good state schools have waiting lists too. So it is difficult, if not impossible, to get into a good state school. Parents move on and places do become available. I was told.
Many good ones are church schools.
You may not like that. Classroom sizes ARE bigger than in the private sector, this is true. This is also something that the state sector is trying to overcome. Nowadays there are school's assistants in classrooms who make a positive contribution towards results. Society has always perceived the state sector as less good, though this seems to be diminishing.