There are ways to work around this. I've just tested this: I've used the URL inspection tool of Google Search Console to fetch a URL from my website, which I've configured to redirect to a paywalled news article. Turns out the crawler follows that redirect and gives me the full source code of the redirected web site, without any paywall.

That's maybe a bit insane to automate at the scale of archive.today, but I figure they do something along the lines of this. It's a perfect imitation of Googlebot because it is literally Googlebot.