Let the state pay the premiums for a year, and then you'll see liability limits come into play. Judges don't get sued for malpractice because they're privileged characters--but judges commit plenty of malpractice that ruins lives. When was the last time a personal injury lawyer tugged at your heartstrings because judges can't be sued for the damage they do? They just shrug.
It isn't just the docs, it's the hospitals. Even nurses are feeling the hit--they're sued more often than you realize.
Same with hospitals and nurses. No care for a patient until the patient agrees to take any malpractice urges before a neutral board. If the neutral board says gross negligence happened, then they're free to go to court. If not, then they're contractually bound NOT to do so.
Insurance companies (auto, house, etc.) have similar clauses in their policies that force you to abide by their arbitration rather than sue them.