This won't be a popular statement, but archivers don't have a legal right to reproduce websites, either. I imagine archive.today is on even shakier ground via bypassing newspaper paywalls. archive.org operates under an opt-out policy which is not how copyright works, and archive.today operates under a "f### you" policy. It's just that website operators haven't chosen to take legal action against this practice.

I understand the innate desire of people to have a "better to ask forgiveness than permission" view of saving old web content (because we all enjoy finding out the thing we thought we lost forever by not backing up ourselves was in fact recoverable), but either copyright law exists or it doesn't. If you want a secure future, make archivers opt-in via a meta tag on pages to indicate the content is under a permissible share-alike license.