Certainly SSN could be improved by making them harder to guess and adding check digits. Other than that, I don't see how any other number scheme is any better. Ultimately there is some number that is associated with you, and is therefore useful for identity theft. Since it has to be used by people and not just computers it'll have to be relatively short and thus easy to steal.

Multiple identifiers per person is a feature, not a bug. People get new SSNs when they want to change their identity (eg, witness protection programs), or because someone stole their old one. That need doesn't go away just because you have a new numbering system.