Personally, I’d say it spends more time in the minor key than the major.
isn't the relative minor just the 6th degree mode of the major?
I just looked up the music again and read that the verse is circle of 5ths counter-clockwise which I’d not noticed myself before, which is neat, you could argue that each ‘movement’ round the circle is its own key - like an Autumn Leafs type arrangement.
5ths counterclockwise is the circle of 4ths...
But. The main thing that pushes it into minor key for me is that it sounds melancholic with occasional glimpses of major key.
Hope this is interesting!
...when you move through the circle of 4ths you draft the next key by flatting the 7th and it becomes the new 4th. no other notes change so the overall vibe of moving through 4ths is a minor sound since you are essentially using a minor fall to get to the next tonal center. Jacob Collier has discussed this.
i'd probably tend to say killing me softly is in G major.