Presumably someone has already built this and I'm just unaware of it, but I've long thought some sort of crowd sourced archival effort via browser extension should exist. I'm not sure how such an extension would avoid archiving privileged data though.
That exists for court documents (RECAP) but I think they didn't have to solve the issue of privilege as PACER publishes unprivileged docs.
In particular, habeas petitions against DHS, and SSA appeals aren’t available online for public inspection: you have to go to a clerk’s office and pay for physical copies. (I think this may have been reasonable given the circumstances in past decades… not so now.)