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Update: I recreated the issue on a staging site for further inspection.
As you can see, pages (start here, homepage, about, contact) still work. So do posts for some reason.
However, archives (for categories, custom taxonomy terms, and custom post types) as well as custom post type posts are all 404.
Weirdly enough, it took a bit to get it fully broken. At first, some archives weren’t 404ing after an update. Then they would 404, but an additional update would fix everything.
Update 2: I think I finally figured it out after a LOT of deactivating plugins, changing plugin settings, and general troubleshooting.
It seems that the plugin I was using to deactivate certain plugins in certain areas of my site (to improve site speed) was causing it. I changed those rules to only deactivate plugins for non-logged-in users and the issue seems to be fixed. Will update further if anything changes.