So as soon as you make SSN secret, companies are going to use another ID to uniquely identify people and we are right back in the same boat.
Identifying a person is one thing, proving you are that person is a completely different concept.
Erm... that's the entire fucking point of Experian. And yet they failed.
Experian is a free-market for-profit corporation.
Assume a credit rating agency can lower a lender's default rate from X% to X'% but introduces an additional Y% of fraudulent loans. If X - X' > Y, that's a good deal for lenders no matter what the absolute size of Y is.
Experian failed, but the system was designed to fail. You can already get all of this important info with this number, so there was little incentive to create a new system and ensure that every American participated in it. Their operation was also entrenched by government credit reporting acts, pretty much setting in stone how things are and preventing innovation. This was a pseudo-government 'industry', not a free market system with new startups frequently joining.
* Government Issued IDs -- Most situations. Police Tickets, Court issues, etc. etc. Usually your driver's license. This is effectively a security token, since its illegal to make illegitimate copies of a Government issued ID (although cryptography should be used in future ID efforts IMO)
* Tax PIN number and Last year's tax refund -- Pay taxes and do tax refunds. There's a ton of OTHER information you need to submit with your taxes to get it processed. The SSN is purely the "username" for your taxes, it isn't the "password".
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In the free market? People use SSN as a password, because free market doesn't give a care about you or decent security practices. Or at least, Experian doesn't (and they're a free market actor)
For individuals, your TIN (Taxpayer Identification Number) defaults to your SSN, and you do not need the previous year's return to file taxes. Not sure where you are coming up with these imaginative claims, and I find it somewhat concerning how confident you feel in bureaucratic design decisions.
The government did NOT pick SSN's for ID. SSN cards used to have 'not to be used for identification purposes'. It was the private sector that appropriated SSN's as ID's. What the government failed in is adequately prohibiting and enforcing a prohibition of this type of use.