Yeah, the comic's story and message are beautiful, but the "sleep kills you" argument is poorly thought out, and based on a shallow understanding of what continuity actually means. It's not about consciousness, it's about continuity. The processes in the brain that make up your mind don't stop as soon as you fall asleep.

There's an argument to be made about how you're never the same person that you were even just a moment ago, because you're constantly changing. That's also shallow and lazy, and ignores the continuity we're talking about.

There's an argument to be made that from your perspective, continuity isn't broken. That's also shallow and lazy, because it treats the perception of continuity as if it's the same thing as real continuity. As far as your clone is concerned, continuity wasn't broken. But I was never worried about whether my clone will die when I go in the teleporter, you know?