To intelligently consider the concept of ingrained sexism for a moment requires some consderation of what sexism actually means. I suspect that many people regard it in the same way as domestic violence/abuse.
With DV they don't really consider it to be Domestic Abuse unless one partner has raised a hand/done physical damage to the other.
With sexism I suspect that many people do not consider sexism has taken place unless someone is openly discriminated against for something due to their sex, like, I don't know, a boss saying that he fired so and so because she was pregnant.
But how about all the times working mums are expected to just drop everything to look after their sick children, whilst working dads aren't. Where a woman feels uncomfortable asking for time off to see their kid in a nativity play, whilst a man gets applauded. Where a woman pushing a buggy gets tutted at, pushed around, generally treated as a nuisance whilst a dad gets greeted by 'aww, how sweet?' type reactions?
Of course there is some ingrained sexism against men - SAHDs, for example, can have a very tough time socialising, being accepted, coping, but this doesn't inhibit their ability to support themselves, to progress in their careers, to live independently or to carry out day to day tasks.
Ingrained sexism is the stuff that people generally don't even question. That a man crying means something is terribly wrong, but a woman crying must be either hormonal or hysterical, for example.
As long as men think it's acceptable to dismiss a woman's emotions simply because she's a woman (and therefore more emotional) then ingrained sexism still exists.