It's certainly not impossible to get a pure interval

Ahh, that's it - I was thinking about the impossibility of getting perfect intervals right the way across a keyboard, as you so eloquently describe, Faldagious One. Yeah, just getting one pure interval so you can hear how it sounds is easy (ish). Mind you, you should hear a guitar I've tuned.

We got a lot of compromises along the way that left us with the idea that different keys have different characters
This interests me. Are you saying that, in a non-well-tempered universe, all keys would sound more or less the same, and have the same "feel"? In other words there would be no advantage to writing music in any particular key (other than, perhaps, to fit in comfortably with vocal ranges).
Edit: ackshully I think musick just answered my question (ta mate)