We already have one. It's called social security and it's a procedurally generated 9 digit number that is vulnerable to series prediction among other things, and essentially can't be (or almost never is) regenerated if it's compromised.

It's a very poor system that was never designed for its current use. We have many of the downsides of a poorly implemented national ID and very few of the upsides. We need an actual purpose built national ID.