In all fairness, if The Godfather is the greatest Catholic movie, it is *only* because of an absence of other great Catholic-centric movies.
However, much of the problem denotes from the history of Italy and the Vatican. The unification of Italy began with the downfall of Napoleon Bonaparte in 1815, and didn’t end until 1870.
Then just about 40 years later, Italy was decimated in a dozen or so horrific battles with Austria-Hungary and Germany, in the Alpine region of the Isonzo River. 650,000 military dead and about 1 million wounded. It had not come close to recovery by World War II.
Yet before, during, and after the war was the peak of Italian migration to the United States. From 1910 to 1920 about 5.3 million Italians came to the US, though about a third eventually returned to Italy.
Importantly, Catholic Italians were greeted by Catholic Germans and Irish, who had been coming to the US since about 1850, so had established a “Catholic infrastructure” in the US that the Italians could greatly expand upon.
In any event, this means that the most profound Catholicism in the movies needs to be based in that period, in the immigrant community, because there you will find not just religious Catholicism but cultural Catholicism as well, integrated throughout the community.
Or because the good father isn't actually looking for such movies. Maybe among Hollywood films that had spectacular success in the secular world it could be the greatest Catholic movie, but if you look at all kinds of films from all over the world it probably isn't.
I feel I should point out that the mafia originated as a Sicilians only club. Not that the very same problems you mentioned didn’t drift over from the boot. Just thought it should be made clear it wasn’t until later that Italians proper were allowed in.