I wonder how much of the desire to hang on to SSN as a private identifier is rooted in the fact that SSN's contain a bit of personal data. Until 1972, the first 3 digits represented where you were born or where the card was issued. Afterwards they represent the mailing address for the application for the card.
Considering the numbering system was created in the 1930s, their idea of what constitutes a unique, private (i.e. hard to remember) number is a bit less relevant today.
IMHO it's just about trying to hold onto it being less generally useful, as was initially promised. The government pulled the same "limited scope" bullshit they did for all the recent "terrorism" "laws".
Enumerating and precisely identifying citizens is the basis of mass surveillance. For most every societal use, a name and a birthday is good enough identification. Individuals should not be perfectly legible to the government, because then the rulers are further tempted into demanding that reality conform to their whims.
Of course now that government-cum-computer has entrenched and precessed, there are many areas where precise identification will help preserve one's rights. If one shared a name and birthdate with a wanted criminal, they certainly would want to be distinguishable!
But such situations are really due to the system asserting that its abstractions are airtight. For example if someone applies for credit in my name, the crime that has occurred is fraud and I am not involved. But the system (within-abstraction) diagnoses the situation as me being a victim of "identity theft", transferring blame to the individuals affected by its own shortcomings! Further cementing the abstractions further strengthens belief in them, making these failure modes even harsher.
Another practical problem in the US is that the government fails to prohibit the use of such identifiers for private purposes, under a half-applied theory of private contract (or really just money talks). So we end up with the double-edged sword of the free market combined with a government mandate, which destroys individuals' sole power of exit versus otherwise unaccountable corporate entities.
And because of that design decision, some states were fairly close to running out of SSNs: https://www.ssa.gov/policy/docs/ssb/v69n2/v69n2p55.html
Today, SSNs are randomized: https://www.ssa.gov/employer/randomization.html
Considering the numbering system was created in the 1930s, their idea of what constitutes a unique, private (i.e. hard to remember) number is a bit less relevant today.