It wouldn't be hard to maintain a hand-crafted database of when domains are reused for something completely different, or even when the same conceptual website has breakages, and use that to choose between Internet Archive or live web accordingly. When one is browsing from an internet archive page, the date is known, when someone is browsing from a live website, heuristics can be used, along with "bisecting" dates when the link is dead.
Ultimately we want more content addressing to avoid this problem entirely (see below), Or DNS -> PubKey, PubKey -> latest content, with some law that the pubkeys shall not be reused for unrelated things. vs DNS which is mere ephemeral Huffman encoding. So see below for the stuff on IPFS. But the trick above is a good stop-gap, and indeed the database itself used to back the extension could be on IPFS.