The rest of your argument around mutual care is reasonable. There are more nuanced views than a blunt "if state support for marriage depends on procreation, let's ban marriage for infertile people as well.", which sounds like spite: if I can't get it, then those guys over there should not get it either because they are similar to me on dimension X.
From the perspective of the state, there is a man, there is a woman, statistically they have a high probability of having and rearing children. Splitting hairs whether they are fertile / infertile is something the state shouldn't get into, partly because it's administrative hell and partly because it's an invasion of privacy.
With gay marriage, there is a man, there is a man, statistically they have close to zero probability of having children, so from the perspective of perpetuating the state, there is no statistically significant benefit in offering them special treatment.
We can now argue whether the state has the right to determine the gender of an individual, or whether that's an invasion of privacy as well. It's pretty easy to tell a man from a woman at birth. Biology and cultural norms make this distinction pretty obvious to anyone who's paying attention. It's both cheap and reasonable to tell apart men from women.