You're confusing civil law (civil suits by the model) with criminal cases (the fed's convicting you for pedophilia). I'm guessing this is a new attempt for the current administration to strike fear in those promoting the first amendment, so expect some significant lawsuits (probably only one or two, but they'll likely get a LOT of negative press).
Generally speaking, the Government doesn't sue. It prosecutes and fines. So no government is likely to file a lawsuit. However, they might decide to prosecute.
But based on past discussions of U.S. Laws and interpretations. I'd have doubts of U.S. LEA involvement with ASSTR unless there happened to be something particularly heinous uncovered by techs while trying to resurrect the ailing server and services, which I admit is possible. (It may have even been the ASSTR Admins who did so in such a case. While hard to imagine, I'm sure there's even a line where Laz might not just simply decline a submission, but refer it to law enforcement)
However, I think their interest would be in whatever that specific content is/was, and the persons tied to that content. Not everything else. Although those other persons might still see a low level of scrutiny by association and opportunity. (Hence advisories to avoid getting associated)
For the Americans it's simple enough as long as they avoided the imagery(and don't partake in that stuff anywhere else). There is no known effort to prosecute for possession or consumption of texts in the US. It is the production(authors) and distribution(server operator) they've gone after historically. Imagery is an entirely different ballgame, and they've been known to try for everyone in such cases. For people outside the US, YMMV and it becomes a question of if InterPol becomes involved and what your own Governments may do.
Most likely use of any data a LEA in the US would make from ASSTR is the potential "breadcrumb trail" it provides in identifying consumers of "pedo" content and seeing what else those people are into that is either flat out illegal or otherwise in a legal grey zone. It basically helps provide circumstantial evidence (probable cause) that illegal activity may be underway and that either further discreet investigation, or an outright search(warrant) is warranted.
That said, at the speed those operations typically move, it could be a year of more before anything happens, assuming a LEA did take it over. IF a LEA is involved, my money would be on one of more pedo rings getting busted open, and prosecution will be over imagery or actual acts. They may throw in "additional charges" as it relates to fictional(ized) stories, as has happened previously, but the focus will be "on the other stuff."