Key , as derived from a functional analysis does have to do with resolution.
Key signature , IMO is of limited use. If you're given a key signature of CMaj / Amin and neither of those chords are present in the music, you're left to perform a functional analysis. In this case, you'll likely come up with E Phrygian. That fits the key sig., so far. What if, you support your resolution to E minor (chord) by using the V7. Now you've indicated Harmonic minor (at least for the duration of the V7) and left your key signature behind.
Interpreting Harmonic and Melodic minor also rely on functional analysis. As soon as you've determined one of these scales "at work", you alter the accidentals in the key signature.
Functioning key and key signature are very different things, IMO
Does this address the point -
It's Friday and I'm a little giddy......
-best,
Mike