Fraudulent charges on a credit card are the least of my concerns. This opens us up to a lifetime of identity theft and insecure accounts of every sort. I'm not even sure how they can approach remedying the problem. Coordinate with the SSA to get 150 million people new SSNs at the least.
There is mo way to even estimate the damage as some devious ways of it harming us may not even exist yet.
Scifi story idea:
Far future. Life extension possible. The government will provide it free (if you want it) - one time only though - when you are near the end of your first life. Upon extension, this technology also turns the clock back to renew you to 20 years old.
You're 78 years old, frail, ready to kick it, but decide to do the extension. You go into the clinic. Give them your information, etc.
We're sorry, you've already been rejuvenated before. We can't help you, unless you want to pay $$$$$$ for us to go ahead with the procedure.
The solution, whatever it is, does not include anyone continuing to pretend that the SSN is now or has ever been suitable for any purposes other than for tracking government benefits managed by the SSA, and possibly also for tax filings with the IRS.
... and all of the other government benefits, programs, or mandated activities, many (all?) of which demand your SSN. Are you even sure that the credit industry, i.e. banks, originally misused SSNs? I wouldn't be surprised if they were required, by the government, to use them, precisely because it is the closest thing to an official "unique identifier".
Some people also might be concerned with not receiving their SS benefits either, which isn't entirely far-fetched given that others might now be using it for nefarious purposes (like trying to collect their SS benefits).
I read something somewhere else (maybe on a different HN thread, maybe here?) that this was changed in 2000 for something called "red flag laws", IIRC.
So yeah - it is required.
There's no such thing as loosing your ssn because it is already public.