It’s funny that you mention Frisk, a character whose gender is explicitly undefined… 
While it makes sense that a story set in a culture with very strict gender roles (e.g. Pride and Prejudice) would be very different for a male protagonist, I can’t really understand why that would be a problem for the other stories. I mean, would the stories really change that much if you switched Lara Croft and Indiana Jones (again, aside from cultural things)?
Can you tell us any CS games which you feel are lessened by the players choice of gender? (I’ll admit that Affairs of the Court is probably one, especially when playing as a gay male, as its gender roles are at once both strictly-defined and all over the place, but do you have any other examples?)