You are looking for Memento.

- http://timetravel.mementoweb.org/ (search seems to be down)

- https://www.webarchive.org.uk/mementos/search (search not responsive for me)

- https://mementoweb.github.io/SiteStory/redirector.html (protocol and tools)

There are a few browser add-ons that will list about a dozen archives for every page you visit. However, that doesn't include non-public services like the one mentioned above.