For anyone curious about this phenomenon, the Ford Foundation is a notable example. Henry Ford was notably racist and the Ford Foundation does a lot that is focused on racial diversity.
>At most 8 years, probably less. And then Yellen will go back to the private sector
Yellen was in public positions from 1977-1980, 1994-1999, and 2004-2018. She was a professor at Harvard from 1971-1976 and at UC Berkeley from 1980-1994. She's certainly made a lot of money from these sorts of speaking fees and there's criticism to be had, but she's spent far more time in academic or public positions than she has "in the private sector".
I don't think she's the same case as, e.g., someone who was employed by an investment bank for a long time, takes a federal position for a year or two, then goes back to working in banking.
https://en.wikipedia.org/wiki/Ford_Foundation