It is an interesting history when you look at both parties from where they started to now.
The Republicans have also always been a Big Government party as have the Democrats.
Now both are as corrupt as they can be and totally useless.
I wouldn’t say always. The concept of big government as we see it now didn’t explode until the 1930s (though the groundwork was laid before then). Real shrinkage of government spending did occur under Harding/Coolidge (the last 2 Republicans who actually managed to do just that). Hoover, sadly, paved the way for the future and (as a Progressive who had been a Democrat as recently as 1920, switching parties only because he knew it would be a losing proposition to be one in that decade) was ill-equipped to deal with the 1929 crash.
If Coolidge had remained in office, the Depression might well have been reduced to a panic (many of which we had had and had recovered from in short order — Harding himself adequately and swiftly handled one at the end of Wilson’s term. Harding is often derided by historians as being a poor President, I consider him to have been amongst the best in the 20th century, economically at least. His downside was having corrupt Cabinet officials because of his trusting nature).
It is curious to note that FDR ran to Hoover’s right on fiscal austerity (a promise swiftly broken). FDR had no real template to follow with respect to the kind of expansionist government (far beyond anything accomplished under Wilson), so he was the trailblazer for the mess we have today. Sadly, after getting an entire generation (20 years) of people indoctrinated into expansion and “liberal” governance, it moved both parties to the left.
Eisenhower should’ve immediately attempted to roll it all back but did not, and virtually every Republican since has paid it only lip service (sincere or not, and that includes Reagan, unfortunately). Limitless government control, spending, et al, moved the Democrats from the austerity and self-control of Cleveland in the 1890s to Progressivism by the 1910s, Socialism by the 1930s and a gradual move towards Communism (that of in practice as opposed to the theory - Marxism is supposed to be lack of government, demonstrating the ultimate hypocrisy of the left).
Although you had two elements in the GOP (just as the Dems did 120 years ago), the Progressives and Conservatives, it was clear that FDR moved them to the left as there were only talk or piecemeal attempts to pull back his leftist agenda (the few remaining elected Republicans in the 1930s after the disastrous routs at the polls were left-leaning). Even as Conservatives made a comeback, they made no attempts (at least successful ones) to launch a drastic rollback to match the successes of the early 1920s.
What passes for Conservative today is much further to the left than decades past. The establishment itself has become Socialist, only interested in preserving its power and has no interest to shrink the size of government at all. Just preserving and managing the overbloated Socialist state the Democrats have enshrined.