Sorry to take so long to answer UQD, I made my post on my way to bed.
"no other kind of pressure group has that degree of influence over the education system." - yes I agree with you. And I am a committed Christian and churchgoer, I should tell you, as I expect it must affect my argument, although I hope it doesn't of course.
The point I was trying to make is that the link between the Church of England and education is a historical one. To save you scrolling back, many church schools were originally charity schools, run by the church, on church land, for the education of the poor which would otherwise have had no education. They were taken over by the state but the church retained, shall we say, an "interest."
So in my view, one of the reasons for the church having this degree of influence is simply historical.
Another reason is that the Church of England is our state religion. A member of our church (like many, struggling financially) is often heard to remark that the Church of England is the only organisation which exists for the benefit of its non-members.
If you want to get married, you are entitled to do so in your Parish Church. If you want to be buried, you are entitled to be buried in your Parich churchyard (supposing there is room). You can do these things if you are an atheist. You are also entitled to have your children baptised, although of course you are unlikely to do so if you are an atheist (but you would be surprised!).
Should tragedy strike your family (God forbid) no matter what religion - if any - you were, you would receive a visit from your parish priest offering support. It is his job to minister to all in his parish, whatever religion they are. This would not be the case for a Catholic priest, or a Muslim, Jewish or Hindu faith leader - they are only required to support "believers." The difference lies in the fact that the CofE is what it says on the tin - The Church of England.
So that's my take on why the Church of England is so present in the education system: because it is part of the state system, and was one of the main building blocks of the education system.
I'd drone on but the DCs are fighting.