One quibble with some parts of your premise: homosexual, bisexual, and transgender men *DO* think about gender on a daily basis. Yes, even those of us who are White.

Why? Because we live in a society that finds ways to challenge our masculinity (and demean our failure to conform to patriarchal variations of it) every moment of every hour. It's certainly a different experience from that endured by cisgender women and transgender women (in terms of unearned privileges we receive that they don't)...but it's by no means a "universally" male experience. At least, not in every facet of life.