Culture isn't going to encourage people to host websites they're no longer interested in, where by "host websites" read "pay money for".
The only solution is some form of legalized archiving. We need the right to copy for archiving without profit, or something. Not sketching out a full, legal solution here, just pointing out that it has to include some form of right-to-archive.
Archive.org, as cool as it is, does not archive the entirety of the internet. Moreover, it's a bit silly to posit that one website that we can surely rely on to exist always! is the solution to the fact that you can never rely on the existence of any particular website in the future.
No, I'm not saying that "the internet archive completely solves this problem", what I'm saying is "there exists no legal barrier to making something like this a reality, since things like the internet archive already exist".
The only solution is some form of legalized archiving. We need the right to copy for archiving without profit, or something. Not sketching out a full, legal solution here, just pointing out that it has to include some form of right-to-archive.