I don't know who exactly (other than the person in question) gets to determine whether or not their privilege "balances" out their oppression. What I do know is that men, and white men, in particular, have vast structural and often hidden advantages over everyone else because for hundreds if not thousands of years, they world was very literally built to advantage them and consider their needs - and in the US this was still overtly true 50 years ago. That doesn't mean that every man or every white man has the same level of privilege with no mitigating oppression or downsides, but they do benefit every single day from that structural inequality whether or not they even want it.

I'm reading a book right now on all the hundreds (if not thousands) of ways that women are disadvantaged simply because even today, an average sized white man is considered the "default" human - for safety equipment, for medical testing, for all sorts of things where the fact that women have different bodies, and different life experiences is never even considered. It's mostly not intentional marginalization but it still has the same effect because for thousands of years, men were "people" who acted in the public domain and women were helpmeets who largely only existed in the private sphere. This has a cascade effect of not just making women less safe, but also less seen - as leaders, as scientists, as worthy of the term "brilliant." This undermines and marginalizes women in ways that you have never once had to deal with in your life ever, even as a not typically "masculine" man. Pretending that somehow gets erased just because someone has wealth or some degree of power is naive, to put it nicely. And the same could probably largely be said of people of color as well. They are not considered the default human in most places in the West and so they are structurally disadvantaged, not just by racism, but by never being considered in the first place. The US has the highest maternal mortality rate in the Western world - by a lot - and Black women have rates that are 3 times that of white women. Some of that is unconscious bias, and some of that comes from being "othered" in other ways. I could go on and on about this but continuing to insist that there are any women in the West who on balance are more privileged than you is completely clueless and more than a little bit misogynistic.

Edit: Read Entitled by Kate Manne and/or Invisible Women by Caroline Criado Perez to begin to educate yourself on just how often and how deeply women are structurally disadvantaged.