One thing that worries me about any form of schooling is "avoiding the biases of the teacher" - how do homeschoolers achieve this?

(It's less of a problem in schools because of the number of teachers, but it's still a problem there).

Well, as you allude to, I don't think there's a general solution; even a bunch of school teachers who are all different people are likely to have come from a group with fairly homogenous views. I suspect some parents who homeschool do it because they can teach to their own biases, while some do it without awareness of the problem, and others actively attempt to find ways to expose their children to things outside their biases. Those in the first and third groups probably think their way is better than the imperfect delivery of government-mandated biases (for better or worse).