Your second sentence builds up two strawmen: 1. That the registry has to be "perfect", whatever that means. It doesn't, it just has to be canonical, and allow for errors in it to be corrected according to some well-defined process. (Not by pulling 20 random documents in front of a judge and suddenly legally become another person.) 2. That these registries are "mythical". It's very much a solved problem. You (I'm assuming you're American) are literally living the only developed country without a registry of who lives in it.
Japan solves this by having the registry in your town of birth, other countries have this registry centralized -- perhaps the U.S. would be best served by state-wide registries, though since migration across state borders is unregulated, I bet that would be very difficult to maintain.
As for the reasons not to have such a registry, I have yet to hear any convincing ones.