Quite right on all those points. Another issue is the phenomenon of “cultural Catholics.” They self-identify as Catholics in polls, and somewhat skew the results. And, yes, the Democrat party didn’t used to be be the liberal party. It became that over the years starting really with Woodrow Wilson.
That’s not to say there aren’t big challenges in the Catholic Church with significant numbers of people voting for liberals; there is. But, it’s gradually changing. The sixties did a lot of damage throughout society, and the Catholic Church was no exception.
Actually, the only three elections that we know for sure that the Catholic vote went republican, is since the 1960s, 1972 being the first one.