I went to state school and I turned out perfectly fine, though I do often wonder if I could have done better academically if I'd had the additional tuition and support that comes with buying your education.
If you got rid of private schools, that just means less places at state schools for students who actually need them, so getting rid of private schools is not the answer. State schools should just be made better.
An earlier poster suggested setting more rigorous standards for people to become teachers but I think it's fine if someone wants to teach with 2:2. Often, their marks in their PGCE's will determine how good of a teacher they are, rather than their marks as undergrads. As long as they know the school curriculum well, 'only' getting a 2:2 in their degree shouldn't be a problem as long as they are an outstanding teacher. I know plenty of people who got 1st class degrees and dropped off their PGCE's because they were a) awful at teaching or b) couldn't handle it. I'd rather have an outstanding teacher with a 2:2 than an adequate teacher with a 1st.