Why did the GOP do so well in early 20th, late 19thc despite massive immigration

[Alcibiades]

To put it simply, Northern Protestants were a majority of the country.

Of course this is a bit of an oversimplification, but the fact that the GOP commanded a demographic majority is very true, because in reality many immigrant groups were not particularly Democratic:

-Non-culturally Southern/non-ancestrally pro-Confederate WASPs (in the literal, broadest sense of the word i.e. not necessarily wealthy) were basically a lock for the GOP (except, crucially, in 1884 when many balked at Blaine’s corruption).

-Germans leaned Republican overall; the Protestant ones strongly, the Catholics more mixed.

-Italian, Poles and other Southern/Eastern European Catholics were mixed, showing strong regional variance in their voting patterns, often in response to local factors; in many cities they were Republican due to resentment of Irish machine control of the Democrats.

-Jews were always left-leaning, but initially split their votes fairly evenly three ways between Democrats, Republicans and Socialists; they voted Democratic at lower than expected rates for similar reasons as Italians and Poles i.e. distrust of Irish machines.

-Scandinavians tended to be Republicans, but also provided the base for Progressive/Farmer-Labor parties in the Upper Midwest.

-African-Americans, in the limited places where they could vote, were staunchly Republican.

This leaves the Democrats with only two ethnic/cultural groups they commanded a clear majority among: Southerners and Irish Catholics. While their unwavering loyalty and structural advantage in the Senate and formidable machine and political organising skills respectively provided the Democrats with an anchor with which to stay afloat and viable during the post-Civil War years, they were nowhere near to commanding an electoral majority, even when parts of the above groups were added. The New Deal changed all this.