To an extent it tracks. No one wants their children to have fewer opportunities than they did, so a ‘rich’ person will probably have to plan on expensive tuitions and other activities. Obviously once you hit the ultra rich this is a rounding error, but for the middle class I see this dynamic at play.

I can afford one private school tuition, I can’t afford 3. Obviously that’s not a requirement for life, but I went to an expensive private school, do I not want my kids to have similar opportunities? Etc.

Who's talking about the rich? Wealtht countries have uneven wealth distributions. The rich certainly can afford housekeepers and nannies and private tutors. A couple of middle school teachers cannot.