What I'd love is a program that automatically downloads and archives any article-form website on which I spend more than a minute in a browser, and makes everything full-text searchable. Does something like that exist? Perhaps it could be solved with a local proxy and some smart heuristics on request timing in a browser-agnostic way.

Probably a little hacky but using a userscript & an archive tool could work. I have a userscript[1] that I've been using for years to archive some news sites but it just uses the Wayback Machine. It can probably be modified to wait more time and send the request to something like ArchiveBox[2] (or any archive tool that has a web interface). The downside is you'd need to whitelist specific sites (or whitelist everything and have a keybind or some content matching to activate the archive function? idk)

[1] https://gist.github.com/antiops/00a37a1de289415fa7cbd9b5d1d2...

[2] ArchiveBox, can have full text search: https://github.com/ArchiveBox/ArchiveBox