This story is pretty revealing about how private schools work. People sometimes assume that you get a much better quality of education at private school, but in practice there's a huge variety in the sector. You have the top schools (Eton, Harrow, Winchester etc) which are presumably OK, but underneath there are an awful lot of schools that are now struggling financially and are pretty clueless about teaching. The main advantage from a parent's point of view is that your child will be mixing with other rich children and have access to good sports facilities etc - but they almost certainly won't have better teachers.
In a state school, something like this would be unlikely to happen. They might recruit someone without a maths qualification to teach maths, but only out of desperation (there's a shortage of maths teachers), not because they thought the maths teacher needed to do the sports coaching. A teacher who had both a maths degree and a teaching qualification would probably be welcomed with open arms.