Man it is wild to read about that election. Coolidge (R) v. John Davis (D) Davis' campaign was spoiled by Robert La Follette running as "Progressive". It was the first election that all Native Americans could vote in.
Davis and La Follette criticized the KKK, very active at the time, while Coolidge, classic republican, didn't say shit about it. The Republicans won huge and controlled the US until 1932. In 1929, the Great Depression kicked off, which probably explains some of the latter discontent.
The Republicans have been very good about making sure a Democrat takes the fall for their economic terrorism ever since though! Truly the more things change, the more they stay the same.
Everett's opinions align well with the Progressives of the early 20th century. Both Republicans and Democrats had progressives amongst themselves at that time, so I bet his vote varied based on the candidates rather than being hardstuck one party.
On the one hand, "a woman can't keep a secret" is blatant misogyny and he deserved to be kicked out for that. On the other, not telling people how you plan to vote (or how you did vote) should be considered normal.