Ya Republicans generally are totally against the SJW nonsense generally. But, when it comes to children there they are. It's still being an SJW.
Just like republicans saying they aren't big spenders and want to shrink the size of the federal government, but then never have seen a military spending bill they didn't want to increase in size. Minus Rand Paul.
I have seen a few just ignore any kind of cuties questions saying they wont take questions on the topic so the opposition to the craziness is probably larger than most people think as you say. Just no one wants to be the guy who says it out loud.