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It might, but the FOIA process might also be the best (or only) process that allows the NSA to practically release the tapes.

Even a bureaucratic process that is a poor fit for a problem at hand can be easier for a bureaucratic organization to handle than an ad-hoc, nonstandard request. Trying through FOIA first certainly seems reasonable.

The NSA has their own museum, has anyone emailed them to ask about this? https://www.nsa.gov/museum/

They seem more likely to be able to deal with such a tricky request.

Find and restore the necessary machine. Do up a blog/website about the project. Share the link on HN. The NSA nerds reading this will find it eventually and get in contact. (Everyone at NoSuchAgency is a total tech nerd. They all know about HN, even moreso than those at NotAnotherStupidAcronym.)