Why would 4chan stop existing? Since when do people stop using websites, "just" because user data was leaked? The hosters can just fix the security issue and move on.
Apparently the source code is full of vulnerabilities severe enough that if they try to put the site back up, it'll be ripped to shreds in an instant.
Fixing everything might not be viable either. There's a reason the site hasn't been updated much in years, they don't have staff to actively work on it.
That's not going to happen for lots of reasons. For a start, no way will the current US govt will spend any more money on bullshit such as "cultural heritage".
Well, there's been clones out there for a few years already. But I don't think, it's going to be quite as business-as-usual, since they can't coordinate a switch to a specific clone, not to mention this clone very likely does not have the server capacity to take on all the users.