During the Progressive Era in the early 1900s wasn’t there a goodly number of liberal Republicans as well as conservative Democrats?
Indeed there were, which was the only reason TR got as far as he did. But liberal Reps alone were too few to win a national election - even a three-way one - and Bryan Dems were more likely to abstain [1] than switch to a Republican.
[1] Or possibly vote Socialist. I don't know what research has been done into Debs' vote, but I suspect that some of it at least probably consisted of Democrats who found the President of Princeton a bit too "establishment" for their taste.