Well, let's see. Sum of divisors is product of 1+p+...+p^k so if that's prime then the number must be a prime power so only one p, and then if the number of terms in that sum is composite you get an obvious nontrivial factor for the sum of divisors, and we're done. I reckon something I can prove in two minutes without making use of the notion of refactorable numbers probably isn't a great argument for the importance of that notion.

Perhaps the point is that this theorem was discovered while thinking about refactorable numbers. I rather doubt it, but in any case the theorem itself seems like a cute curiosity rather than something any number theorist would care much about.

(I am a mathematician, though I've been out of academia for years and was never a number theorist.)