WASHINGTON -- A group called the Herndon Alliance -- a coalition of liberal health-care groups, unions and patient-advocacy groups created in late 2005 -- was only a few months into its work planning a health-insurance overhaul by the time it asked focus groups what they thought of the idea of a government-run plan to compete with private ones. The public-option was an article of faith for many in the alliance, but the focus groups' reactions were sobering. Skepticism ran high. The chief worry: Giving access to inexpensive government insurance to America's 46 million uninsured would boost costs, or reduce care,...