For what it's worth, my marriage was the same situation as yours(and my now ex-wife was just like yours). Not seeing eye to eye on simple things like this caused too many fights. I want my child to have up to date vaccines and be in public school, and I need my wife to have a decent job because of her lifestyle demands, etc.
After a long debate, we made the choice to move into a good school district and send our kid to school. This was a long fight, as she had family in rural Oregon who homeschool all of their kids, and based on the stories of the kids(dumb as bricks, pregnant by 16, etc), I did not want to send my kid down the same road. She also had separation anxiety(it was bad enough that she did not spend a day apart from him until he was 10[sleepovers] or travel without him until he was 13).
I went to private school for elementary and public school for the rest, and I was very satisfied with my experience, and I've made friends I otherwise would not have made. My time in private school prepared me academically and I was able to do extremely well in public school(entered a year ahead in some subjects), which was beneficial because doing well in public school in California gets you a great deal in state aid to state schools as well as guaranteed enrollment in state schools.
I don't think there is a replacement for the social education school brings(and public school was much more educational in that regard than private school), but I don't think that that is the only reason why. I also think that school provides the opportunities to advance just as much as if you were a real teacher homeschooling your kids.