They became a defacto identifier for many institutions just because US citizens and permanent residents generally have one - but aside from convenience, they're pretty problematic compared even to something like a UUID/GUID.
If you were trying to design a true national ID number, you'd probably want to approach the problem more like a database designer would: you'd want something that had possible unique values far exceeding the reasonable number of people you expect to exist before the collapse of society, that was reasonably random and difficult to predict based on other known attributes about the person, that by spec was never shared with another person, etc.
In any case, the private key is stored on a plastic ID, only released with a PIN, and the databases only store the corresponding public key.
A leak of a public key without the private key is harmless.
When I go to a bank for a loan, I should give them some form of private id as proof I am who I say I am (this doesn't have to be a federally issued number).
When companies communicate about me with each other (such as running a credit check on me which is likely why T-Mobile even had SSNs) they should use my public id.
The solution is pretty simple, the government should require others to prove they engaged in a transition with you before being able to put an unpaid debt on your credit report, not the other way around.
You appear to be operating under a misunderstanding: The government doesn't organize credit reports. Credit reports (in the US, at least) are compiled by independent companies, who operate with very little oversight or recourse (and occasionally leak lots of data themselves).
We already have such a system, it’s called a passport, and it has complementary “cousins” like passport cards and nexus cards. That framework could easily be built upon, but there is a mountain of paranoia and stupid to climb first.
Instead we have the current system where fraud is endemic and where millions of people are marginalized by poor access to ID.
In any case, the private key is stored on a plastic ID, only released with a PIN, and the databases only store the corresponding public key.
A leak of a public key without the private key is harmless.
SSH been using it all these years and still going strong.
There is literally no reason why Americans would trust the central government to be more secure than T-Mobile at this point.
After all, how did the NSA's top-secret Vault7 weapons get stolen considering that they were following "best practices"? And then proceed to bring havoc to the world with WannaCry...
If anything it might encourage companies to smarten up if there was a law on the books that required companies that get compromised to pay for the re-issuing of the user tokens to every impacted individual.