How old are your other children?
I don't think it's good that a six year-old is critical of the authority figures in his life. It's too cynical at his age (not that I don't understand and agree with him!). So I'd be inclined to give up on the school. Your son sounds more advanced than my kids and I know my kids could learn all they need in an hour of two a day of instruction (althought they are avid independent readers, so there is a difference there). I doubt the school is currently adding much to what he is already learning at home.
If you are willing to homeschool, but just don't want your life to revolve around it I am sure there are ways to do this. Why not allocate the tuition money to an au pair or full-time nanny and home school? The private school option will always be there.
With regards to him being loud - I have trouble tolerating this and have made it a priority to encourage my kids not to get loud in the house. I think it's OK to impose this, especially as it makes us all enjoy each other's company more.
