So if that's the case, I think we should move beyond really even trying to think of this info as private or a marker of identity, and we need to move everyone to more secure forms of identity verification.
As has been pointed out on HN before, "identity theft" is a made-up concept to make it seem as if you had something stolen from you, when the real problem is banks and other service providers do an absolute shit job of identity verification. They're the ones at fault, and they try to shift the onus onto you to fix things when they screw up.
Indeed, a social security number is pretty much the only additional piece of data to the stuff above that one would need to open up a bank account in someone else's name, and those have been leaked plenty of times too.
The government needs to make harsher penalties for banks and others that can ruin your credit, etc. because they accept all this leaked info as "proof" of identity.