Which means we no longer need to store and handle them securely, right? Can I have yours?

Mine was leaked several times. At least once by the government itself. See https://iapp.org/news/a/21-5-million-breached-in-second-opm-... for example.

SSNs where never meant to be secret. It's an ID not a password. You can thank banks and credit card companies for treating them like a verification system.