At least with an insurance company, I can dump them for another if they provide poor service, or sue them. If the government runs it, I will be stuck with whatever the politicians push on me. Has anyone tried to sue the government here? Private insurance companies are the lesser of the two evils. Companies may deny claims, but governments deny proceedures. As I understand it, if you are over 52 in Britain, you will not get dialysis or a transplant if you are having severe kidney problems. Oregon instituted limitations on care about 10 years ago.
Having worked in insurance as a broker, I can verify that they do stamp "denied" all over everything. Usually because their claims department is short staffed and this is the quickest way for the clerks to clear their desks.
What comes as a surprise to most people is that insurance is not a gamble at all from the company's perspective. Using actuary tables and their own history, they can usually predict within 1 or 2 percentage points what their claims are going to be for the upcoming year. Same with employers that provide health insurance for their employees.