Just look at the longer passage that I quoted in the OP. It directly deals with some ramifications if you hold that the collapse of the wavefunction is real, namely spooky action at a distance becomes an even worse problem.

There is more than one interpretation of the passage you quoted, and I think a more neutral interpretation is more likely. In the first paragraph they highlight interpretations with collapse, explore the implications for interpretations with collapse, and point out an unintuitive consequence: "This is especially mysterious when two entangled systems are at separate locations, and measurement of one leads to an instantaneous collapse of the quantum state of the other."

In the second paragraph they highlight interpretations without collapse, explore the implications for interpretations without collapse, and point out an unintuitive consequence: "But if the quantum state is a physical property of the system and apparatus, it is hard to avoid the conclusion that each marcoscopically different component has a direct counterpart in reality."

The paragraphs have the same structure. Ramifications are mentioned for both cases. So when you say "I also don't see how you can claim that Many Worlds is unrelated when the linked paper itself mentions ramifications for that case," it seems to me like you're reading asymmetrically. The authors appear to have simply explored the implications of their theorem for interpretations with and without collapse - and of course there were unintuitive bits for both, because it's quantum mechanics.