I think the notion that it has to be "men are privileged" or "women are privileged" is not a helpful way of thinking of things because it makes things an us versus them situation, and it washes out all the nuances and distinctions.

I think a better way to look at it is that there are hundreds or thousands of ways small and larges that different genders (and races, and sexualities) have privileges over others, that these go both ways.

So, to pick a random example: A man is privileged in the sense of being less likely to be raped during his lifetime, assuming he avoids jail. But among rape victims, women are privileged in terms of their access to supports, sympathy, and so forth.

These also interact with wealth, with race, with sexuality, and so forth. The male norms of being a "provider" may work fine for someone who is affluent, but very poorly for someone who struggles to feed themselves. Certainly at the poorest levels we see a disproportionate number of men who are homeless, and the evidence suggests this is because of greater supports for women at those levels.

The "we must pick a privileged gender instead of picking apart the hundreds of different forms of privilege that apply in all directions" thing really tends to lead to "why care about men's issues, anyway?"