On most instruments, a key signature merely indicates how notes on the staff should be interpreted when they are encountered. On some others, however, a key signature may indicate how an instrument should be set up. Indeed, on some of the latter instruments, switching between an altered and unaltered pitch may take long enough that playing accidentals would be essentially impossible unless they are set up before the performance begins.
When writing for instruments of the former kind, there is a very strong convention that pieces which are in Lydian and Mixolydian modes use the same key signature as the corresponding major key (Ionian mode) and add accidentals, and likewise that those in Dorian or Phrygian modes use the same key signature as the corresponding minor key (Aeolian mode), also adding accidentals. If a piece like "The Little Drummer Boy" (Mixolydian mode) is rooted on G, it would have F naturals but not F sharps. Nonetheless, most performers would find it easier to read with an F sharp in the key signature and natural accidentals before each F, than it would be without the sharp in the key signature and accidentals on the notes.
When writing for instruments of the latter kind, however, what is most important is ensuring that the performer correctly sets up the instrument before the performance begins. If a performer will have to change to switch a string between F sharp and F natural during a performance, including an F sharp in the key signature and F naturals throughout the score would be disastrous if it caused the performer to configure the instrument to play F sharps. Even if the performer configured the F strings to play naturals, littering the score with F natural accidentals would serve no purpose since the performer who encounters F's in the score would have no chance to play F sharps whether or not the notes were marked.
If a piece has no F notes at all, then it would make sense to play with the instrument configured for C major if the preceding and following pieces are also in that key, or for G major if the preceding or following pieces are in that key. If editor would expect the piece to be performed between two other pieces that are in C major, adding F sharps to the key signature would require at minimum adding a note at the start of the "G major" piece indicating that there is no need to change the instrument's tuning. While that might be workable, it may be easier to simply omit the key change.