Ireland has a PPS number which everyone who’s ever had a job or claimed government benefits has. The union of those sets is basically everyone. Clearly that hasn’t helped.

That is far from everyone.

Well.. yea, because it's the US that creates the list, and they don't know or understand that systems like ID numbers exists.

Americans use their social security numbers like id numbers in all sorts of systems. This problem has nothing to do with id numbers.