If that was all it (SSN) was used for, a "unique-id", it would work ok for that usage.
The problem is that far too many companies also make use of SSN as a "secret only you know" to authenticate that you are in fact the individual identified by the SSN. I.e, your "login name" is identical to your "password". It is this miss-use that leads to the problems around SSNs.
OK, fine, whatever, except this is the same even for controlled substances. Which is quite weird considering all the other regulations around controlled substances. One time I had to get rid of some excess oxycodone after a surgery, I ended up having to go into a little booth at the sheriff's department where they had a drop box and a sheriff's deputy sitting at a desk behind a window watching me. But all my wife needed to pick it up in the first place was my name and DOB.
[1] - https://www.pcworld.com/article/3004654/a-tale-of-two-women-...
No. It's sufficient for identifying someone, but not at all sufficient (not even close) for verifying someone is who they say they are.
SSN as a user identifier - to uniquely describe an entity
Actual PKI including a secure public/private key with a signature from a trusted agent (like a government ID authority), and maybe also web-of-trust signatures? THAT is what is required to securely sign things.