[icarus_drowning]

How is "tell[ing] you what key the piece is in" different that defining tonic/dominant/leading tone relationships? Those relationships are the product of the whole/half step pattern that the key signature communicates. It isn't just important that G has 1 sharp, and it is F. That sharp actually defines the leading tone.

[colomon]

That's only if the tonic is G. I play tunes all the time with one sharp, but a tonic of D. (ie D mixolydian.)

[icarus_drowning]

Sure, but I'd argue that's because the key signature is wrong. The correct way to notate that is with either a modal key signature a la Bartok or notate it as D with a constant natural on the C, showing that the tonal center is D while clearly notating the departure from the traditional major scale.