Because the Democratic Party is a big tent, coalition party. Republicans are much more homogeneous from an ideological and demographic perspective.
This was once true, but is not really true now.
I'm not really sure what you think has changed. Could you be more specific?
One cannot deviate from the Leftist line on Social Issues. Everything is tilted to the left. Democratic liberals who are pro-life and not conforming to the social issue line are marginalized in the party. They are not acceptable as Presidential candidates.
How do you think a pro-choice Republican would do in a primary?
Outside of a purely local race, very poorly. "Regular Churchgoers" are a significant constituency in the GOP primary and they have networks that inform those folks as to positions on social issues regularly.
It would probably depend on region, right? The Northeast might be different from the South here.
Even in the Northeast, "Regular Churchgoers" would be Catholics that are likely in agreement with the Vatican's position on abortion.
When I was a young activist, we were looking over the election enrollment books, district by district, and I noticed one address that had all Democrats in this heavily Republican district. It was explained that this was a residence for priests and nuns and they were all Democrats. This was, of course, a time where there were a number of pro-life liberals; the Massachusetts delegation was almost all liberal Democrats, all but one a Catholic, and all of the Catholics, regardless of party, were pro-life. Nowadays, these folks are Republican; the Democratic Party has refuted them on their most fundamental issue. The Catholics that vote Democratic are lapsed Catholics and marginal Catholics that identify as Catholics, but who are not regular Mass attenders.