If you have a grunge voice, then by default you shouldn't call your voice a metal voice, and Staley's voice wasn't really even as diverse as some other grunge singers.

They can tour with whoever they want, or call themselves whatever they want, genres are a very small/petty aspect of music at the end of the day. But the drumming is very grunge, the bass doesn't matter (unless he's playing some old school funk and I missed it), and although the guitar has metal traces, it doesn't do enough to classify the whole band as a metal band.