It doesn't even need to be government-run, we just need the right incentives. I've seen proposals for making some kind of data loss insurance mandatory to compensate victims. The insurance companies would then conduct audits which determine the premiums for the company, and investigate for negligence after a breach.

Edit: Thinking more about it, this would probably also be positive for security investigators. If a company is stonewalling you and ignoring a legitimate bug report, you now have the option to escalate this to the insurer. Maybe they could even facilitate bug bounty programs for smaller companies