What would you do instead of SSN? Or how would you change the process of providing a number for income tax purposes?
I have no ideas for implementation, I just think we're having the wrong conversation.
As for what I’d do instead: create an application that is fully managed by the IRS and let me login with some combination of a client certificate, password, U2F device, TOTP, et. al. Have the IRS securely implement something like OAuth (I hear “securely” doesn’t quite fit with “OAuth” but they could implement something custom-built if they want to) so I can get the IRS to tell whoever I need them to tell that I am me.
Ideally, replace IRS with a purposeful identification organization for these purposes and let the IRS use what this new organization gives them.
We could easily avoid this mess of PII honeypots by repealing the federal income tax. The country was not founded with a claim on your labor.