The SSA should just set a time limit, (let’s say 3 years,) and then publish a database of every single name, DOB, and SSN. Force the banks to figure out a new system of identification, by making the current system useless.
The current system is already insecure; SSNs were never intended to be secure. So why has the SSA tolerated this for so long? Just make the “in three years we’ll publish this live database for anyone to search” announcement, so banks are forced to develop a better system. It gives them the time to work on a new system, eliminates the need to keep SSNs secret, and the SSA can keep operating as normal.