Other than being government and wanting to spend new money, they would have been better off taking the online Medicare/Medicare Advantage program and modifying it. It works.
The Medicare/Medicare Advantage website provides for county and state variations in the Medicare program, and it has been active, with annual modifications, for at least half-a-decade.
I would think a large percentage of the uninsured are people between jobs. Covering them could have been a simple case of having people on unemployment insurance automatically being eligible for Medicaid, and giving them their Medicaid card when they apply for unemployment. But that would not have justified a massive increase in government.