Because private health-care systems do not have to serve everybody, they can serve the people who have bought in much faster than public health-care systems can. This is both convenient and occasionally life-saving.
By encouraging competition between insurance companies, coverage prices can be kept down and health-care employees' salaries can be kept up as companies compete with one another for customers and doctors.
A private health-care system encourages innovation as companies try to cut costs by creating more efficient ways to deliver health care. These savings are passed onto the users.