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 present an IP issue.
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.
Can you describe this in more detail?
This tells Archive.org's crawler to immediately process a page and add it to the Wayback Machine's cache. Unfortunately there's no public API for this, but it is possible to programmatically submit a request to their endpoint and scrape out the resulting archive link (and I have code that does this, if that would help).