The installation process at the time would have made him suicidal.

Cobain died in April 1994.

The first Slackware Linux release was in July 1993. Debian Linux¹ 0.9 came out in January 1994; SuSE Linux launched in March 1994.

So yes, Kurt could have used Linux.

But he probably would have had to install it from floppies.

¹ It wasn’t called Debian GNU/Linux until later; see the 1994 revision of the Debian Linux Manifesto.