Actually, the Democrat who ran against Coolidge was much more of a conservative. I can’t remember his name now, but he was against racial integration and was for low taxes as well.
Harding was definitely more liberal than Cox, who was also a bit of a reactionary. Hoover was part of the progressive, liberal wing of the GOP. FDR ran to Hoover’s right in 1932, and didn’t become full blown liberal until 1936.