It seems like there should be a service for the interesting documents to go to historians or something. Then they can take their time going through it to see what has historical value and what does not, and make a non-split-second decision.
Another option would be to auction entire hard disks on Ebay with a description as per the above, not sure if this already exists. Then it's a kind of lottery, with massive potential payload for people willing to go through the data.
If that technology existed and was good enough, it would be an interim solution. Grab the data, pulp the paper.
Really, given how cheap and easy it is to store large amounts of information digitally I'm surprised that something like this isn't common for a lot of cases to reduce storage costs.
How many businesses have rooms full of accounting records and receipts going back decades that could instead be taking up a bit of space in a rack? Toss a copy in Amazon Glacier for a penny a gigabyte so you have something off-site and call it a day.