Definitely public for Primary school.
After that, it depends on the kids and what the father thinks as well, I suppose. I wouldn't send them to private school if I didn't have the money, but if I were very rich I probably wouldn't care much either way. I have a preference for co-ed public schools because it puts students in touch with reality, if the kids plan to join in with reality later on. But, high performing schools would also prepare them for a university environment better, IMO, assuming they are going to uni. If they're bright, I'd encourage them to sit scholarship tests and try out for select entry schools.
I really don't have a problem with sending the kids to the school down the road (provided it has a nice vibe and actually seems to care about the students and the students aren't stabbing each other and so on) but I don't mind sending them to the prestigious academy either.
I think I like small schools (or at least, schools which cultivate the sense of smallness). More individual attention from all staff that way, hopefully by the end of their stay at school most of the staff would actually know who they are. Small class sizes are also a must.