They don't always use it, because I can archive a new page from my mobile phone browser, which doesn't even support extensions.

My guess is that most content providers with paywalls serve the entire content, so search engines can pick it up, and then use scripts to raise the paywall - archive.is takes their snapshot before that happens / doesn't trigger those scripts.

There are known ways to bypass paywall which are just impossible to implement within a browser extension while trivial on 12ft or archive. For example, to use Ukrainian residential proxy as some news websites granted free access from.