That is possible, but in smaller communities where everyone knows each other, I see teachers and parents working together to help their kids succeed. I don't want to discount the value of a good teacher, but I think it is very unfair measure someone's teaching ability based on their ability to overcome poor parenting.
As far as the important piece of paper, I think that is something of a myth as far as college admissions go. I've heard a lot of parents say that they need their kids to graduate from a public school so they can go to college.
All the college admission requirements I've seen are the same for public school graduates and kids who graduated from a non-accredited program. It mostly comes down to your ACT and SAT score.