Military, sure, but driver’s licenses are state-level, not federal. Health care has been using birthdate like a password (one that is largely publicly available) for way too long now. At least financial institutions can use account numbers and financial history and code words, but even all that isn’t great.
It’s a messy patchwork, but I think at the time of the creation of the SSA, the US may have still thought of itself as a land of second chances. IBM numbering Holocaust victims probably didn’t help the idea of a national ID, nor did the victim narrative of groups like the NRA.
I’m not sure if it’s possible not to have a national ID anymore, so denial of it just forces a terribly kludgy implementation from whatever is around.