Usually the phrase "voice of a generation" is used metaphorically for the words someone delivers in text or song rather than for their literal vocal skills. Kurt Cobain earned the tag, but so did F. Scott Fitzgerald, a guy never known as a singer. But if you take the expression literally, it's hard to find a Gen-Xer more deserving of the title than Chris Cornell.
Even if you aren't a huge fan of Soundgarden (and I'm not, always preferring Mudhoney's grunge Stooges to Soundgarden's grunge Led Zeppelin) or Chris's other work, there is no denying the power of his voice. Many have noted his four-octave range, but he also had impressive intonation and emotionally delivery. Yes, he could scream and wail, but he could also offset it with quiet contrast. Mariah Carey could be a another claimant of the title, and you don't have to be into her musical choices to appreciate her talent.
Chris Cornell wasn't just the frontman of a one of the biggest bands that exemplified Generation X. He was one of our best voices.