There shouldn't be any fee paying schools. Nevermind more!
A couple of other points. The state has slashed the amount of regulation it imposes upon schools in many ways - look at the Academy stuff. Of course it'll still chuck a few whims down here and there, British Values being a relevant one. But the overall trend is away
As for faith schools, I doubt they're all the same. My daughter goes to one and the only thing of note thus far is she knows a few Jesus stories. When she's older she can decide to reject faith, like me or retain it, like her mum.
But that's her decision to make.
Finally all schools are involved in ideology (Althusser blah blah blah), being secular (either in the British or the French sense) doesn't touch this.