I guess I can understand the concept of "equaling out unfair burdens" in the scenarios where it's a heterosexual relationship -- especially if an established couple is trying to get pregnant and/or the wife/girlfriend is already carrying their child.
I still don't see what that has to do with everybody else outside of their relationship.
Men don't menstruate. Yes, that's biology. If a man was bleeding from his penis, that would seem to indicate that there was something BIOLOGICALLY HARMFUL going on with his body.
Now, if you're saying a heterosexual man who does extra gestures of kindness for the woman in whom he's interested (while they're still in the courtship phase) might be signaling to her what sort of a husband/father he could be (once they're raising children together) -- yes, I could intellectually understand how that's attractive to a woman if he is someone with whom she is seriously considering having children.
But when you exit the realm of heterosexual child-bearing couples -- any logic attached to that gender-specific behavior really goes out the window.
There's no reason, for example, why a random gay man should be expected to automatically hold the door open for a random lesbian woman just because he has a penis and she has a vagina.