That's true. It's hard to admit that, though. The current monied class can easily blame previous, (mostly dead) generations for racism and misogyny and say "we didn't create this problem, but this is what we're trying to do to fix this". Of course, they're benefitting from the situation and remain in control per the status quo.
If we admit to the classism that currently exists, that in the US the zip code you were born in says so much about what your opportunities will be, it's going to take a lot more social reform to address it than applying pressure around university admissions or job offers. The monied class may actually have to give up some power in that situation...
It's darkly amusing to me how many well off leftie progressive types are on board when I talk about difficulties that revolve around being a lesbian or disabled but look horrified when I drop into working class speech patterns or refuse to deny the class aspect of my upbringing.