Happily, some of the questions you ruminate on have been addressed experimentally by the moral psychologist Jonathan Haidt. "Liberty", as he calls it, is the last of the six "moral tastes" that he outlines in The Righteous Mind (2002).

Importantly, however, he also provides good evidence for five other moral tastes which are important to people of different political persuasions to different degrees; these help explain some (but not all) of the conflicts that you talk about.

Unfortunately, I think they also largely prevent everyone from rallying around freedom to a very great extent. For instance, the freedom to do things that are viewed as disgusting is unlikely to be motivating to someone who has a strong moral sense of purity.

(If the coronavirus response had been framed as a purity issue rather than a care/harm issue, I bet we could have seen exactly the opposite response: the right instituting mask mandates while the left screamed about individual liberty.)

But I think that there is still a lot of room to maneuver even if we don't explicitly acknowledge different moral instincts, and yet more room if we do.