Identity theft is possible for the sole reason that privacy advocates consistently defeat proposals to build any system of identification/authentication less broken than forgeable drivers licenses and shared-secret SSNs. People actually get angry when we make it harder to forge IDs! Propose any sort of PKI or .gov OAuth provider or something, and you're basically Big Brother himself. It's like the ability to commit identity theft is considered a right.

Trying to authenticate people by their knowledge of data that's in or easily deduced from the public record (or "shared secrets" shared with every entity they've ever done business with) is an objectively wrong security design and we need to stop using it. Making those public records harder to access is an absurdly circuitous fix to something other than the real problem.