Maybe add a link to 'cached' in addition to 'web'? You could either do your own caching when the link is submitted, or use a service. The link will be useful immediately if the site goes down over load, and keeping an archive will keep the comments comprehensible for future readers if/when the link erodes.
I really like this idea from a historical preservation perspective. HN comments contain a lot of a value in aggregate, and it would be a shame to lose necessary context to exploit this value due to simple link-rot. However, I can see issues arising with paywalled links. The HN cache would likely display a rather useless paywall for many of the most popular stories. Navigating around the paywall by technical means may…
Archive.org does follow some rules (e.g. robots.txt and passwords as you mentioned), so not everything would be guaranteed.
It could be a good start for a lot of the content here, though.
EDIT: Forgot to add that I'm not sure how much it'd help with sites going down from the HN attention. I could easily see archive.org moving a bit slower than HN's users.