Yes - but they still pay providers better than does the government, since insurance companies can't use armed force or coercion and cannot force costs to be moved to others.
Insurance companies also relentlessly pursue efficiency since a more efficient competitor can take their accounts by offering lower premiums. Government has no such drive.
It all sounds idyllic until you consider that insurance companies, insurance brokers, medical billings specialists, and businesses' benefits plan managers are all middlemen who are heavily responsible for driving up the costs of health care.
This notion that a proper, rational government cannot pursue efficiency is a dogma, btw. Perhaps the current corrupt American government might not be able to do it, but that doesn't mean that a rational government could not.
Of course, any way you slice it, the fact remains other nations are spending far less on health care as a percentage of GDP (while often having more doctors per capita) while having 100% coverage than is the U.S.