I’ve said for a while that the SSA should do basically this exact thing. In a more controlled manner, but still the same result. Announce something like “in two years, we’ll make our database public. Every single name, DOB, and SSN will be publicly searchable.
It sounds radical, but SSNs were never meant to be a secure form of ID. Old cards even said something like “do not use this as ID” on them. But organizations quickly latched onto it because they wanted to have a way to identify individuals with the same name and DOB. And SSNs were convenient because people already had them.
It would force organizations to develop their own way to ID people. It would be a huge step towards making an actual secure form of ID. And the warning time would give people enough time to design the new system and roll it out, while still giving a hard deadline for when it needs to be done.