If I pay Fat Tony to break your legs, and then you pay a doctor to put them back together, this whole adventure increased GDP, but is a net-negative-sum activity.
"People have to do more work to accomplish the same amount of results" isn't a worthy goal in itself.
It's not. If I steal a million dollars from you, the net impact on GDP is ~0. Hell, if you were keeping it in your mattress, and I spend it on drugs and parties, GDP will actually go up.
GDP is a useful number, but it's not the only useful number.
Could one carefully construct examples, sure, does that scale - not so much.
Over in Manitoba they've got one of those for car insurance too. Amazingly, people continue driving there just the same, the government insurance plan never needs subsidies and Manitobans pay way less for insurance. Almost as if the concept of the free market being the best solution for market X needs to be tested once in a while just to be sure, and results may be surprising.
The larger point being: there are many areas of the economy in which the free market approach is clearly better, and there are many other areas of the economy in which the free market approach is clearly deficient and a government-run agency is superior. Insurance is in neither of these convenient buckets. "Unlikely" is an opinion on a hypothetical; it is not necessary when real-world examples abound.
In terms of this exact comparison - is this like for like? I don't understand it well enough to know.
(Not to mention suboptimal health outcomes arising from the perverse incentives it creates.)
Only if the net value is positive-sum.
Setting a city on fire, or giving cancer to a million people, or just going down the street, smashing every parked car's window with a tire iron will likely increase national GDP.
Liekwise, people pay a lot of money for net negative-sum industries - the health insurance industry in the US immediately springs to mind. We'd all be better off without that parasite.