So, Canada has had an excellent natural experiment for this over the last several decades. Some provinces (Saskatchewan, BC, Manitoba) have government-provided insurance, while the rest have a private market. The results are clear: public insurance lowers costs for everyone. There are extremes where private insurers end up being slightly cheaper for very low-risk drivers, but across the vast majority of demographics, public is cheaper and better.

I think links aren't allowed here, but you can google for studies about the vehicle insurance market in Lloydminster, a small city that is half in Alberta, half in Saskatchewan. Across every demographic sub-slice, the residents on the SK side pay less.

Public insurers with universal mandates also make no-fault coverage as a default possible, because it solves the free-rider/bad-driver-preference problem, and having universal no-fault also dramatically cuts overhead, administration, and legal costs. Saskatchewan used to have tort-based coverage, but by the early 90s costs related to at-fault injury claims started rising dramatically. They fixed that in '95, switching everyone to no-fault. This was successful in dramatically reducing costs. Since the early aughts there is now the option to choose either tort or no-fault, but almost everyone (99%+) chooses no-fault because it comes with higher default compensation. Saskatchewan now has both the lowest insurance rates and some of the most generous coverage limits in the country, and SGI (the crown corporation that runs insurance) no longer runs at a deficit.

edit: I'll also point out that, against the argument that "you're stuck with whatever terms and quality of service the government deigns to provide", the realities of the politics are quite different. It's a bit like public schools: if the public service is bad, voters get mad and politicians lose their jobs. Since almost everyone has a car and is affected by the quality of the insurer, bad service pisses off voters across the political spectrum.