It seems to me a parallel path that should be pursued is to make the impact less damaging. Don't assume that things like birth dates, names, addresses, phone numbers, emails, SSNs, etc are private. Shut down the avenues that people use to "steal identities".
I hate the term stealing identity, because it implies the victim made some mistake to allow it to happen. When what really happened is the company was lazy to verify that the person they're doing business with is actually who they say they are. The onus and liability should be on the company involved. If a bank gives a loan to you under my name, it should be their problem, not mine. It would go away practically overnight as a problem if that were changed. Companies would be strict about verifying people, because otherwise they'd lose money. Incentives align.
Identify theft is not the only issue with data leaks / breaches, but it seems one of the more tractable.