Maybe live in an area with a good public school and send them there?

I don't really want to live in an area where public schools are comparable to private schools. In the US, those areas are often suburbs and I prefer living in the city.

I also think your kids will have the same experience in a private school as public.

I don't know enough people who went to suburban public schools so I can't really compare public schools in the poshest suburbs to private schools, but you may be correct in terms of the social experience.

I do think that the peer group is going to be stronger at a good private school though because they try to award scholarships to students who wouldn't otherwise be able to afford tuition. In contrast, public schools in posh suburbs will only have kids from said posh suburb. It's not like the public school is going to bus in a poor, smart kid from the city because their math team coach wants the kid at the school or whatever.