I think that differs quite a bit by country. At least in Denmark it's administered in a considerably different manner, more like any other public spending that decides on priorities and allocates resources. Some parts of it do have actuarial elements that parallel what an insurance company would do (pooling risks across the population), but many don't. The cost of many things is well known in advance and has little uncertainty (especially when the whole country is taken in aggregate), and it's just a public-health choice whether to pay for it or not. Routine childhood vaccinations are a clear example, which are just budgeted as a public-health measure, like water purification or whatever. Hospitals have an element of risk-pooling when individual patients are looked at, but at the public level are run more like fire departments or police departments, which also have an element of risk pooling. I guess you could argue that fire departments are really some kind of state-administered fire-response-insurance system, but I'm not aware of that being a common approach. Data and statistics and people to look at them are still needed though, of course, but they are looking at more "real" problems, like which procedures best improve morbidity-adjusted life expectancy.

I know in some other countries, like Germany, it is set up much more closely to insurance, in the sense that procedures have prices, are billed out to patients, who are covered by insurance plans that reimburse patients for costs, etc., in which case I imagine the bureaucracy is much more insurance-like.

The dynamics of the financial incentives involved are a bit different with goverment provided health care and private clinics funded through medical insurance.

Private clinics which are the main receivers of insurance money are often owned by the doctors. Thus, they have an incentive to over-charge, over diagnose and run expensive labs whenever they can. I'm not saying it happens, just that the incentive is there. Government provided healthcare offers no financial incentives for practicioners to over-diagnose - on the contrary this system then incentivizes to be as prudish as possible with the public resources.

I'm not saying one or the other is better or worse, just that they are not equal in every way.