"Should the internet be archived?"This would be a great question to ask researchers in archaeology, anthropology, linguistics, history, etc.
My own impression is that the more data there is, the better for researchers. They can always weed out what they don't need later, but there's no way to get back information that has been destroyed or lost.
Also, in the past people have made some horrible choices regarding what's valuable and what's not. For all sorts of reasons, from political to psychological to social, they've done things like burn, discard, or destroy texts and artwork we now consider valuable but they did not. Often, what was considered valuable (like sacred texts) was not really as revealing about their daily lives as things they did not value and discarded.
So I'd really think twice before we discard any records. If there's some kind of serious pressing need to be selective (such as lack of space) that's one thing. But if keeping it more would not be a huge burden, I'm much more of an inclusionist than a deletionist.