However, your point still stands true in the case of custody rights, since same-sex-marriage opponents tend to claim (rather dubiously) that children raised in such a family tend to experience psychological trauma.
I'm really more-or-less neutral on the matter; it doesn't matter what it's called so long as the rights are equal for everyone. "Marriage" is a good-enough term for that.
Some folks express that, yeah. I do not believe it is an honest position in the general case and have suspected since the jump that that's a move of the goalposts designed to appear more reasonable than they are, ready to be dragged back further as they are approached.
Because its not coded language designed to appeal to homophobes, unlike actual Republican rhetoric. If it wasn't an "attack on marriage" it would be something else.