The top Trump administration health-care official said Dec. 4 that allowing people to buy into Medicare plans would cause the “complete and utter destruction of the employer-sponsored health insurance system.”

The “public option,” a government plan that would exist side-by-side with private insurance, is effectively just another iteration of a proposal to extend Medicare eligibility to everyone, Health and Human Services Secretary Alex Azar said a during his remarks at an American Enterprise Institute event. The “Medicare for All” idea...