Rather than being about fast/simple/cheap, I think using SSN as a key was more about the fact that SSN is the only common identifier that almost all US citizens have.
I think you're using the word "key" differently than OP. You're talking about identifiers, and they're talking about security.
SSNs were a good potential identifier, until the people that needed security cheaped out and started using SSNs as a bad implementation of security. Now they're bad at both purposes!
Tax ids were never meant to be used as a form of global identification. If you go look in a real bank core, you'll find this field does not have any uniqueness constraints.