I'm only an amateur, but I doubt there are string players that "learn" equal temperament. I have no idea how I would find 440 * (2^(1/12) ^ n) Hz, for any n not a multiple of 12, in the way that I can find 440 * (4/3) Hz, or 440 * (3/2) Hz, etc. When playing with equaled tempered instruments like piano, you just listen for clashes and adjust dynamically, which is only going to happen in slower, sustained passages.

And you're right, we don't play "based entirely on just intervals." What we do is constantly adjust our intonation depending on whether we need it to be "just" with respect to something else (like other notes in a chord), or whether we are free to use a more "melodic" intonation. See https://www.youtube.com/watch?v=QaYOwIIvgHg for a good demonstration -- note that he talks in formal terms like "play x in the Pythagorean system," but I think you can largely see this as a rationalization of what players do naturally).

Finally, the presence of vibrato doesn't really obviate intonation concerns, sadly. There's a lot of theoretical debate about how the pitch of a vibrated note is perceived (is it the highest pitch in the range that determines whether the note sounds in tune? etc.), but in practice you can easily verify that adding vibrato to an out-of-tune scale will not make it sound any more in tune, nor will adding it to a shift mask a slightly-missed shift (if only!).