They can't shut down the service because there is no single service. They can go after individual servers, and that is fine. The admin is responsible for what happens on their server.
Why would an admin run a server if they might have the feds knocking at their door? Sure, people might run nodes that host only their content, but they sure are disincentivized from running any sort of shared service.
This is the same problem as TOR exit nodes and why 3 letter agencies run most of them.
This is the same problem as why an admin would run a forum if the feds might knock at the door, or why an admin would run a website about cats if the feds might knock at the door. It's an overblown fear.
People do host those things, so clearly they don't need convincing. I'm replying to you because you are the one who said it on Hacker News, and I'm concerned that people might read the FUD on Hacker News and it might stop them from hosting something.
Isn't the design of peertube that you can view videos originally uploaded by other instances? So unless you decide to not federate, you are responsible for every video on every instance.
Yes, but you can federate selectively, as in choosing which other instances you want to. And, yes, federating turns you responsible for the content of the other instance (IMHO; not legal advice or even fact checked).
Someone is still hosting the content and paying for resources just like with every other service distributed or not so it doesn't really add anything new to the equation.