"Are British parents prepared to be 24-7 role models for their children?"
Well, I should hope so.
No competent and caring parent should expect schools (or any other institution) to instill values in their kids.
Schools taught my sons the 3Rs, I taught them right and wrong. In doing so, I not only reared good kids, I also contributed to the strength of their schools.
La-de-da parents (regardless of ethnic or economic class) are the Number 1 problem for public education, the underlying cause of societal ills that have invaded the classroom.
You are correct. However, I expect the schools to enforce common rules of civility which they don't.