"The point is that there is underfunding, and objections to faith schools can be made on highly ideological grounds (parents should not be able to have their children educated in a particular faith), but otherwise make no sense."
No - that's simply not true.
What makes a school desirable (particularly with secondary schools) is not just good management and a stable staff, but the children and the parents use it.
The comprehensive where my daughter is likely to end up if she doesn't pass the entrance exams to the selective schools in the borough is actually quite well-managed and has a reasonably stable staff. However, nobody I know wants their children to go there because it is full of children from the most deprived and difficult parts of the borough. The selective system concentrates the most disadvantaged children into one place and this impacts on classroom management and the overall running of the school.
Secondary schools with very large numbers of children from disadvantaged backgrounds generally get worse results than schools who have a middle-class intake, no matter how well managed. They end up near the bottom of the local league tables and get into a spiral of self-perpetuating failure. It's very hard. They don't attract the children from motivated, educated families, who understandibly want their children to go somewhere where many children are achieving highly. They end up with disproportionate numbers of difficult children which puts huge pressure on the staff, which results in high staff turn-over, staff absence etc, which in turn damages the education of the children.