The statement in the OP is kind of stereotypical towards gay men and lesbians. I don't think either of them are particularly more happy. I think it's just the fact that women are expected to do this whole settling down thing since they are children while men have more freedom towards dating but considering the fact that it is kind of hard for gay men (and lesbians tbh) to find someone due to discrimination and such, it's not surprising that they would have a harder time settling into a relationship.
That's a whole lot of massive generalizations you've got there
I know gay men who have been in domestic monogamous relationships for a decade and I know lesbians who bring home a different person every weekend.