Can this search browser history? I've seen browserparrot[0] for this purpose but it's a bit abandoned from what I can tell.
If I recall correctly, Firefox keeps your browser history in sqlite, seems trivial to add your own "doctype" in order to support it, as it's open source (https://framagit.org/medoc92/recoll) and written in a modular way (check the pdf handler as an example: https://www.lesbonscomptes.com/recoll/usermanual/usermanual....)
If you have something doing this to every page you visit, and Recoll can see it, then Recoll can index it.
Regarding automatically saving every page you visit, there's multiple tools that do this. One I played with and liked - but I can't remember the name - it's 5 numbers and refers to a port you can type with localhost to search through all recorded pages. That or something like it would work really well with Recoll.