I know, I meant he was a small-l (right-)libertarian ideologically rather than a big L Libertarian in the sense of actually being a member of that party (which IIRC wouldn't even exist for another five decades or so). Man opposed Prohibition, high taxes, agricultural subsidies, excessive economic regulations, police unions, the KKK and the League of Nations - he was probably the closest the US has had to having a genuinely libertarian (ideologically/philosophically) president since Grover Cleveland. Since the Democrats post-Cleveland had largely become the party of centralism & regulation (and would logically be even more into that shit in the Southern Victory timeline where the American political spectrum was dragged much further left with a successful Socialist Party) I really don't think he'd want anything to do with them, even setting his historical Republican partisanship aside.
Coolidge also makes people who fetishize an all-powerful central federal gov't seethe to this very day, so that makes him even more based as far as I'm concerned.