Public insurance is not in all provinces. I know BC, Saskatchewan, Manitoba, and Quebec (partial). All the other provinces don't have public insurance.
It was originally started (in Quebec, at least) due to the foot dragging by insurance companies regarding the payment of claims, and in Quebec, bodily injury claims specifically. The idea was to have all the drivers in the province insured, so no possibility of uninsured motorists. The original intent was great. What has happened over the years is that the public insurance organizations, while still providing for faster claims settlement, are not great for non-normal (e.g. modified, special interest, antique, etc,) type cars. In addition, instead of making insurance more affordable even for normal, daily driver type cars, it has gotten more expensive. In Quebec, there was a small decrease in private, property insurance, but the increase in registrations fees (from about $40 to $250, probably higher now), and drivers's license fees (used to be $65 for 3 years, its now $85 for two years) lead to an overall increase in costs for everybody. BC, Manitoba, and Saskatchewan run full public insurance, so they not only cover bodily injuries, but property damage (including the car itself) as well.
Collector car insurance is available in Alberta from Hagerty, Silver Wheels, and although I haven't seen it, probably Grundy as well. Same limitations, no driving to work, but no monthly limitations. Insurance companies here also specify no "shopping", which means no shopping centers.