"Managing" supervisa insurance is called insurance supervision and is done mostly by provinces, but is also done as part of financial oversight (partly at federal level). While doing more in this area specifically on supervisa insurance would obviously require some resources, it is something government does relatively routinely - makes sure that insurance policies offered are not complete rip-offs (consumer protection) and that the insurance companies can make good on their offerings (financial stability and various other rubrics).
It's not magic, and neither is it free; but neither does it have to be government "providing" this insurance, (micro)managing product offerings, or putting public money to risk, etc.
Oversight of insurance offerings by setting minimum standards in particular areas and licensing companies that offer them is pretty bread-and-butter stuff for insurance market regulation. (Complicated in Canada due to federal/provincial responsibilities etc., but not insurmountable).
That said, I'm still skeptical any govt would want to take it on (coordinating various levels of govt and navigating overlapping bureaucracies and related issues across 10 provinces is just a lot of hassle for not that many people), but let's not pretend that it would necessarily require government devoting vast resources. They could mostly just name a product that's required, set coverage standards, some min/max coverage, and let private insurance companies figure out the rest (although yes, there would be lots of policy issues). It's not the supervision that's expensive and resource intensive, it's the coverage.
The real problem (IMO) is likely to be that the type of coverage people basically want/expect for a supervisa is simply going to be expensive. Leading back to the problem of PGP overall, if it's not limited by a non-income quota, it would have to be limited by actual cost, which is unpopular - the old 'selling visas' complaint.
So most likely is some kind of lottery, whether called that or not.