"We'd love it if the file-system could detect this and try correcting first, and if it couldn't correct the problem, it could trigger the restoration. But that only seems to be an option for RAID type systems, where the drives are colocated."
If you have ~2TB of irreplaceable memories set up a NAS with a RAID array. whilst bit-rot can be detected it can only correct itself if the file system knows what the bits should have been. To this end BTRFS and my recommendation ZFS can be set to say scan all data 1 a week/month etc and using the redundant data in the RAID array correct the 'Bit-Rot'.
I have a intel atom board in a old case with 4 drives(2x 500GB mirror and 2x 1TB mirror). I have FREENAS on this it is powered on every night by wake on lan. Backs up any new data and gets shut down. once a week it backs up new data then runs the command 'zfs scrub' this checks for bit-rot or inconsistencies in the file-system and corrects them if any are found.(can email you a warning if you want as well). This way if any files get damaged on a home pc/ laptop etc.. any user can turn on the NAS and recover there files from the shared folder.
1 point of warning ZFS is RAM hungry so 4GB is the minimum. something to keep in mind when ebaying for a old pc to use. others will also point out that file transfers are ~20-30MB/s with a low powered atom so use something with more grunt if its to be a 24/7 NAS.