The by far best system I've witnessed was the Swiss healthcare system, which is NOT a single-payer system.
Some features: (1) health insurance is obligatory for all residents (2) must be private, cannot be purchased/sponsored by company (3) minimum coverage is specified by law (4) health insurers are private companies, often (mostly? always?) non-profit (5) they cannot reject applicants, and can only discriminate (by price) on: (a) age, (b) residence (i.e. more expensive city/area => more expensive health insurance) (6) all procedures are paid - 10% copay is mandatory (up to a certain yearly amount) (7) health insurers make extra money on better health insurance ("private coverage") offering better service, more experienced doctors, private hospital rooms, extra coverage (e.g. for mental health, abroad etc.) etc. - those can discriminate on much more features, including existing health conditions sex (e.g. for young women it's more expensive, because of pregnancy)