As the person who runs usesthis.com, let me tell you it's a massive pain in the ass (I have a JSON file for every item I link to, and so many just... die.)
Have you considered automatically submitting the links to the archive.org save API? Unfortunately it doesn't support HEAD anymore, only GET but basically just request this URL:
Have you considered linking to a Wayback Machine archive page instead?
It's funny, I was seriously considering doing just that a day or so ago. I think I might.
You could have a service that runs in the background (say once a week) on all outbound links that checks for 404s and then automatically updates the outbound link to the archive.org link. That or have both links.