Look, can you find any substantiated example of a Republican pushing for an equal draft with democrats opposing it?That was 2022. 2021 had it's own round of that, contemporaneous to the court case and which had no chance of passing at all. This year had another round of it, if I recall. So did 2016.
Often it's a republican pushing it when Dems are in power, because if it passes it can be used as election fodder to their benefit. "The other guy wants your daughters to come home in body bags" and the like.
Because I can only find examples of the opposite

Congress moves toward requiring women to register for the draft - Roll Call
Congress is moving toward requiring women to register for the draft after the House included a requirement that they do so in a defense bill.
Like, nobody's pushing hard on it, due to it's completely irrelevant nature, but whenever it comes up it's only some of the GOP that oppose it