This is a weird way of looking at it. String players aren't sitting there consciously thinking of their tuning as they play - they're doing it by ear in real-time. The tuning they use will be the one that best harmonizes with the other notes being played at that moment.

> And it sounds off for music that doesn't largely sit in a single key signature anyway,

That's actually where the ability to adapt tuning dynamically is the most powerful - it allows you to be in tune relative to other pitches being played in that moment, not just in tune relative to some absolute benchmark that nobody is going to be able to hear anyway (because almost nobody has perfect absolute pitch).

Well, that's your problem. You're using a keyboard, which doesn't permit you to harmonize dynamically the way an unfretted string instrument does.

Even within a particular key, the pitch that sounds the best for a particular note will depend on which other notes within that key you're attempting to harmonize with. A keyboard can't do that.

"Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.[4]"