This system hasn't lasted ~90 years because they just throw someone in a chair and let them figure it out on the job.

Any reliable system, electro-mechanical or digital, needs thorough user training and checks.

The worry with this one is it's a single authoritative record with no easy way to backup or replicate it. They say there are non-authoritative (at least legally) digital versions of most(?) of the records. I hope/assume they're actually more consistent with that than the video makes it seem because those are the only feasible off-site backups they really have. If not one fire is all it would take to wipe out an entire countries SSA program.

This is a government office. A government should be able to build the technical knowledge required to keep a private signing key secure.

I do agree that individual-to-individual cryptography is more difficult, but how often do you need to check the authenticity of a document from a friend or acquaintance, digital or otherwise?