Then we got income taxes and federally guaranteed pensions (Social Security) and medical assistance (Medicare/Medicaid) and a few other, smaller safety net programs. Furthermore, while there's plenty of fighting around the margins, the overwhelming majority of Americans are broadly in favor of keeping at least some of the Great Society programs, and that means we have to have a national registry of some sort.
It turns out that when the population is broadly distrustful of the idea of a national registry but broadly in support of a tax regime and social programs that require a national registry to function the stable equilibrium is a really really shitty national registry that everyone hates but nobody feels they can improve without getting shouted out of office. American politics is dumb.
Why do they want a NUMBER? Because names are too... human. They're messy, and they can change, and they don't fit standard forms. It's an impulse to purge the humanity from social identity that motivates establishment of national ID numbers.