Well, bud, it helps to make it clear that you're talking about people when talking about American politics if you actually use "the people on the Left" or "Lefties" (and similarly to the Right) rather than "the left" and "the right".

Sadly "the left" and "the right" in American politics are commonly used to mean Democrats and Republicans, when none of them is actually Left-of-center politically, probably not even anywhere near the center of the Left-Right axis, though on the Liberal-Authoritarian axis it's a different story as their main differences are along that axis.

My point still stands that Moral Liberalism is hardly an exclusive of the Left, though when it is on the Left it arises naturally from the broader fight for Equality, whilst on the Right it is at times (like in the case of the US) just a way to display a strong political difference from the other party roughly in the same place on the Left-Right axis, which is the point the OP was making.

(In countries with more parties that the American power duopoly this is often more obvious because you end up seeing parties which call themselves Liberal and are both Rightwing on Economic Subjects and Morally Liberal alongside parties which are both Leftwing* on Economic Subject and yet just as Morally Liberal as the "Liberals")

So, no, in the US context not all of the right "are trying to deny trans people rights" - half of them are and the other half want equal oppression for all non-elites, trans or otherwise.