I agree that that statement is technically correct, but I have observed various forms of it in American media, and here's my issue with it: the Conservative platform seems to not have run on it at all.
Broadly, I don't see the point in straw manning (conflating all immigration with illegal immigration, for example) in a good faith discussion. I've seen it a huge amount in left wing media, but I think honestly have to dismiss that discussion as bad faith and try and desperately search for the remainder.
Specifically, they haven't run on "gay Americans" that I've seen at all (I'm not sure what that would look like, even), and they haven't run on green card holders. They have run on trans ideation and surgery for children, and trans women in female spaces, and illegal immigration, it's true, but that is far more specific, and it's those precise issues that got them elected.