If each has its own system, what happens if someone form one province needs care in another? Does the one providing care bill the one where the patient resides?
You have an insurance account with a Health Insurer who happens to be owned by the Province you reside in. Even if you are out of Canada, your province of residence Public Insurer pays for care. You are advised to get supplemental private travel insurance, but if I get bonked on the head in Chicago, I present my Provincial Health Insurance card to the local hospital, they bill the Insurer just like they would bill any insurer.
What would be different for me if I were to be in Chicago would be filling out of forms. We don't do that in Canada, you just present your Health Insurance card. The process isn't different if you are out-of-province, but whom gets billed does.
As a resident of Saskatchewan we also have supplemental insurance for motor vehicle injuries (everyone is covered in North America as a passenger or operator) which provides things like income replacement and travel cost reimbursement, and then a third layer of insurance if you are a driver as your Driver's License also carries an Injury Insurance coverage component. Finally there is conventional plate insurance, No-Fault or Tort (you elect which you prefer),comprehensive all-peril is mandatory but as the Insurance Agency is non-profit, not expensive, relatively. No difference in cost due to age, sex, etc, just actual rate of accident and actual average cost of repair for the make and model you drive.
In other words, as Commie as you can get, and the epitomy of Evil to the private insurance industry.
Auto insurance became a thing in the late 1940's when the Ontario-based auto insurance companies refused to offer policies to rural residents of the Province. The government said, ok, you don't want us, we'll self-insure.
There is nothing particularly special about Health Insurance or any other coverage, it's an insurance company like any other. The only difference is who pays the premiums, but when coverage is required to be payed out, it's exactly the same as if you were with State Farm.
Cost savings are found because there is only one Schedule for coverage. An Aspirin from the hospital dispensary is on line 302 (or whatever) for every patient. With Private Health insurance, you need a staffer at the hospital somewhere to fill out the individual company's insurance reimbursement form, because every Schedule is different with every Insurer. A Canadian hospital will have few administration staff.
At St Paul's where I last visited, there are six Insurance Administration secretaries for 500 patients. My Doctor is at a clinic with 15 other Doctors, there are two reception / administration staff who handle all billing (and everything else, including patient scheduling) for all 16 practitioners.
FYI: I have been a victim of an auto accident while in the US. I have first-hand experience with how it works there. The forms, the forms ...
But like I say, I live in the Province that the Federal Health Act was modeled after, we have deluxe health care and better than other provinces. A buddy who now lives in BC told me on the phone about a year ago, when we were talking about the time he was injured on the job and was out of work for two years, " If I hadn't been in Saskatchewan, I'd be begging on some street corner right now" (he earns a six-figure salary) which is a bit of a stretch, but is a bit illustrative as well.