> It doesn't matter if men are being left out of jobs (statistically) — they're [according to the ideology] the eternal benefactors of invisible, omnipresent systemic privilege.
This implies a contradiction that doesn't exist.
Centuries ago, the aristocracy was statistically left out of jobs and also the eternal benefactors of (quite visible!) omnipresent systemic privilege.
There are multiple potential reasons for men to, statistically, be taking fewer of a set of newly created roles. It could certainly be some systemic bias against them, but it could easily also be that they are choosing not to take them for what are surely good reasons for themselves. It could be that fewer men are interested in new jobs right now period, relative to women. I'm sure there are many other potential explanations as well.