"Catholic social teaching" is best addressed by the European right wing (organicism, integralism, distributism, social credit, national syndicalism, corporatism, and ultimately fascism/falangism). Because the American right wing is relentlessly capitalist and (classical) liberal, the American Church had no one else to turn to but the Democrats.
The notorious "right wing extremist" Father Charles Coughlin (perhaps the first politicized preacher) began as a supporter of FDR and the New Deal (even going so far as to say "the new deal is chr*st's deal") before turning against it for not going far enough. He also advocated the nationalization of the financial sector, the railroads, and key industries. How did this guy ever get to be considered a "right winger?" By associating capitalism with the Jews and being anti-Semitic (though he later supported Israel as a "bulwark against Communism," something most American conservative politicians of the time were not willing to do). It is this same logic that labels Huey P. Long and Dr. Townshend as "right wingers."
With regard to the coextensiveness of "church" and state, one could argue that in the Divinely-ordained and regulated government of Biblical Israel this distinction did not exist (even though there was a "division of powers" between the king, the priesthood,and the sages in the Sanhedrion). Similarly, tithing for the purpose of caring for the poor (among other things) was a Divine Law just like "thou shalt not kill." What both Lockeans and socialists fail to grasp is that in this case, legally-mandated provision for the poor was but one aspect of a Theocracy. The removing of this duty from religion to rationalism/pragmatism is one of the tragedies of mankind's plummet through history.
Centuries of fuedalism have left their mark. Many people are still looking for the lord of the manor to take care of them. Socialism is just a modern form of fuedalism wearing a mask of concern for the common man.
"Catholic social teaching" is best addressed by the European right wing (organicism, integralism, distributism, social credit, national syndicalism, corporatism, and ultimately fascism/falangism). Because the American right wing is relentlessly capitalist and (classical) liberal, the American Church had no one else to turn to but the Democrats.
Whatever "Catholic social teaching" meant in the 19th century, when it was associated with traditional monarchies, and whatever it meant in the early 20th century transitional period, when all political, social, and economic structures in Europe were shaky and everyone in Europe was going to extremes, since WWII it's generally been associated with the Christian Democratic parties, parties which tried to synthesize political democracy and what remained of traditional morality with the new institutions of the welfare state. Whatever that synthesis would be in American terms, in Europe faced with Communism and militant socialism it was a conservative project.
In postwar Germany and Italy, Christian Democratic parties have generally been the conservative parties. Not entirely, of course, there are a few small "liberal" parties that used to combine free market philosophy with middle class values, but in Germany, as in Britain, the liberal party (the Free Democrats in Germany) have tended to become more "liberal" in the American sense. Italy's Christian Democrats have pretty much collapsed since the end of the Cold War, and the remnants have joined the opposing left or right coalitions.
As for America, c'mon, really. You know about Prohibition. You know about immigration restriction. You probably know about Al Smith. You may know something of the public school wars of the 19th century. You know that the evangelicals of the day -- the Methodists and other denominations that have since become "mainstream" -- were the backbone of the Whigs and then the Republicans, and that they didn't much like the Irish or the Catholics or alcohol or festive German Sundays or even dancing. You know all that, but you take this down a totally different -- and much less justified -- road.
Now I agree that there's something in a pure free market ideology that doesn't fit in with Catholic social teaching. You might reflect on how well it fits in with the teaching of other religions. I don't know how things are now, but in the 60s, Catholics were less inclined to be ideological free marketeers, but also less inclined to be ideological socialists. It cut both ways. Maybe it was because American Catholics still felt a bit foreign here or maybe it was because they didn't have the idea that a single text or single idea could dictate what should be done in practice. But in any case the reasons why Catholics leaned to the Democrats for so long are quite different from what you want to believe now.
I gather you were a Catholic for a while and moved on. These aren't things you might have heard in church, but "cradle Catholics" -- whatever their current religious or political beliefs -- are aware of them.