The hypothetical system that makes most sense to me for this: a process that runs 24/7 on a server, watching your feeds on those services. Grabbing and saving everything via APIs or screen-scraping.
Yes. Why do you call it 'creepy'? It's supposed to be a personal service, owned and controlled by the user who wants to archive stuff.
And all computations consume resources, and may have bugs. So what? They can be optimised. Bugs can get fixed. The process would, ideally, auto-update.