However, wouldn't having a public option be more efficient? If the
private sector has to compete with a public option, then private
companies would be driven by market forces to perform better or close.
If the public option is truly better, then consumers would naturally
buy into it.
Not really. The public option would likely be cheaper coverage in a like-for-like comparison. Medicare already has lower administrative costs than private insurers do ($132 per person vs $700), so it's already more "efficient," and a public option would be more efficient than private insurance for the same reasons. The competition would never help to make the public option any better, because its price already are set without a profit motive. The competition might make the private insurance cheaper, but it will never be as cheap as the public option until they are operating at 0 profit (and even then it won't be as cheap, because they have greater administrative costs).
Meanwhile, an existing competition between public and private insurance would be keeping the public option from having all of the market share. The problem with this is that, the more people are covered by a public option, the more that economies of scale can help lower administrative costs per customer, because there will be fewer bureaucrats working per customer. For example, the people who write software that helps price medical procedures will not substantially balloon as more customers come on board. 20 software engineers can write a program that serves 2 million customers as effectively as they can serve 300 million customers. So there are certain kinds of bureaucratic administrative costs that don't grow with more customers, and that means more customers results in proportionally lower administrative costs per customer.
Would abolishing private health insurance be the method that gets most
Americans the best coverage? What are the economics behind that?
Private insurance is analogous to a flat tax, and single-payer healthcare is analogous to a progressive tax. This is because most people who have a single-payer plan have in mind that the burden to pay for it would be disproportionately shouldered by the wealthy. Private insurance, on the other hand, places an equal absolute burden on a person who makes $10,000/year and a person who makes $1 billion/year. So, single-payer healthcare is likely to help ensure that all Americans have health coverage, no Americans are crippled by extraordinary medical bills, and wealth inequality would be eased slightly. Private insurance would continue to keep tens of millions of Americans uninsured, many Americans will crippling medical bills that ruin lives, and continue the tide of wealth inequality that is steadily growing in America.
Related economic consequence of single-payer: this would also free-up more spending money for lower and middle class Americans, who are the classes that proportionally spend the most of their money, and spend it the fastest when they get it. This would lead to increased consumer spending, increased economic velocity, increased job creation in the private sector, lower unemployment, increased wages (and probably some inflation too). Having more Americans covered under healthcare and fewer Americans bankrupt due to medical bills also implies increased quality of life.