Well, I think most people, including white men are oppressed unless they are filthy rich. That doesn't mean that they don't have privileges (more than the others).

And from a practical perspective, toxic masculinity limit them denying them opportunities. Sure they refuse to learn skills that are considered "girly" because that way they can try to claim privileges, but by rejecting those skills, they become less than they could. And in the medium-long term that never is good.

They fail to identify that the problem is the same "traditional" values that also hurt women.

Of course they are scared, they gave up on some things (feminine traits and skills) in the hopes to get others (male privilege). If they lose male privilege, they would be at a disadvantage (which is what is happening now) so they try to go back in time and return to a past where male privilege make up for lack of talent, hard work or skills.

They are victims of their own prejudices, not of feminism or other Social Justice movements. And I think it would be good to point that out and that the way for them to stop being victims is by rejecting patriarchal values and putting into value or those skills and talents that they have been neglecting (but that are available to them too).