But they require large support systems (typically rich parents) to work for "free". Scholarships mean they're not free.

This is indeed how it works in some countries, but depending on how strong a nations social net that isn't necessarily true.

Social net is a fun term, it's just earned value reallocation.

How many lavishly-paid deans and bureaucrats and administrators are employed for each grad student?

It's not easy to disentangle and it varies by country and institution, but those positions are not normally directly funded by public research grants.

They are, because every university (in the US) takes anywhere from 10-50% off the top of every grant

Generally universities take half the grant from the start and that's before taking fringe on top of salaries.