Thanks for your comments.
Looking at the source code for this experiment, I think it would be pretty easy to exclude the query string part of the URL, which often contains tracking IDs (google analytics, for instance), and sometimes carries session IDs. I’ve opened a bug to suggest this to the archive developer team: https://github.com/internetarchive/FirefoxNoMore404s/issues/45
The problem I could see with a manual URL check is that, in many cases, there might not be a matching page in the wayback machine. Seems like a worse user experience in general.
Note also that 404s aren’t sent from private browsing pages, so that’s an option you could try.
Thanks again for the feedback