It's just silly to assume that Tyson meant that the entire universe doesn't care about you.

It is usually silly to assume that Tyson means anything particularly deep, when he speaks outside his field.
He does that a lot, and it is usually toe-curlingly cringe.

@David, I agree that Neil probably means as you say but those weren't quite his words. The real summary is likely that Neil agrees with Norm's comment (although he never replied to it). And that Norm's comment is a nice reminder for everyone. The universe does care.