I think we might have different ideas of the role of biology in our lives, but, you are absolutely right that gender differences manifest culturally. That's not to say they wouldn't exist without culture. Culture and expectations shape the way we think and behave, so there is nothing wrong with using cultural cues to shape a character. There's nothing disingenuous about it either, because it's how human beings shape their own identities. So if you say "I can't think of any gender differences that don't come down to culture," I wouldn't say you're inattentive at all, I would say you're looking too hard.
When I make characters, I take cues from real people around me. If these are cultural cues, it makes no difference. I look at my male friends and I see competition under a veneer of civility, I see conflicts that get resolved instead of simmering, I see easy bonds of friendship but private worlds that are never discussed. When I work outdoors, I see men returning to work before women, taking greater risks, and offering help instead of requesting it. When I work in the office, I see women preparing, instructing, and laying the tedious footwork that must always be done. In a workplace populated by women I see seething tensions and secret alliances, in any environment.
I see women who pass by donation boxes but cringe at the idea of a stray cat suffering, men who cannot help but pretend to be experts at everything, and I see little cues that make a character. The things that make a person interesting. They don't have to be true for everyone, they just have to evoke the right kind of person.