This is especially so under Ike, who people think tag-teamed with Rocky to basically crush anyone who wasn't part of the old NE establishment. It's a nice story, but it's not really true.
Ike was very popular afterward, but he did not sweep the GOP primary in 1952. He instead ran neck and neck with Old Righty Robert Taft and had to play all sorts of backroom deals to get a majority of delegates.
He spent much of the primary "playing to the base" in a way that Goldwater fans would like and at least rhetorically was pro-smaller government.
Even so, many R Senators and Congressmen were not Ike men, and they often gave him trouble or killed reform ideas he had. National politics is not just the President.
It's true that the NE establishment was very powerful (the only real bastion to survive and endure the FDR sweep, but it was weakening already then. There's a reason Rocky said "I'm all that's left" of it in 1964.
We also forget that while Ike did expand Social Security, he was a genuine budget hawk, was quite cautious in foreign affairs and intervention. His efforts to keep the defense establishment trim was not out of pacifism but to make sure it did not overwhelm the country.
*in the sense of budget needs and such.
Another important fact is Ike was def. a driver of religious revival in the 1950s, adding "under God" to the pledge of allegiance and starting the National Prayer Breakfast.
He had his flaws and drawbacks, too. He was too slow to move against McCarthy and his insinuations against Truman as too soft on the Cold War were absurd. He was also a little *too* cautious on civil rights sometimes, tho not always.
We also forget that the GOP was already moving in a more conservative direction by his second term, pushing right to work. Twas really bad timing - union membership was at an all-time high and there had just been a short but sharp recession - and the GOP was hammered.
But already by the late 1950s and 1960, the more conservative parts of the right were reasserting themselves under Goldwater (after Taft had passed). They did not all suddenly drop dead or bow to Rocky, and most came from outside the South at this time.
IRL, the wing that dominated in Ike's time was the rough middle ground, and Ike was the linchpin. The liberal and conservative wings waxed and waned but never died. Same under Nixon.
All this rambling for a core point: The two parties are coalitions. Always were, always will be. No one wing remains entirely dominant for long. Tis a matter of historical circumstances, social realities, and even sometimes the refutation of ideas from time to time.
If the Ds do have one advantage over Rs - and it's a big one - it's that they generally share the same overarching goals and mostly disagree on tactics and how to distribute the spoils. Rs, being all about principles good and bad, often deeply disagree on goals.
Left hatred is more than enough for it to be an opposition and to win elections...but that lack of real concensus hampers governance. The best Rs were able to form ad hoc coalitions of the wings based on shared strategy.