Coolidge + Harding were conservatives and in favor of racial equality. Wilson, was a progressive and in favor of segregation.
Unfortunately history disagrees with you.
You said "always." The Democrats of the 1910s and 1920s were no one's heroes and had their base in the south. Woodrow Wilson is pretty much a unique figure in history. I would not disagree that for many periods of time, the Democrats were worse on racial equality.
What did Coolidge and Hoover do, affirmatively, for racial equality? Not much.
By definition, being conservative meant preserving the status quo, and for most of history, the status quo was segregation and racism. The exceptions were abolitionism and Reconstruction (Republicans, 1850s-1870s) and the slow move toward civil rights which began quietly in the 1930s and 1940s (when FDR warred with the conservative wing of his party over anti-lynching laws) through to the Civil Rights Act of 1964 and beyond. Liberal Republicans often fought for civil rights and conservative Dems against prior to the ideological sorting out of the parties.
