You presuppose that American insurance companies are not a very effective cartel, primarily functioning in the regulatory capture industry, with a side business in insurance.
It would be cool if what you describe actually happened, but in the real world, premiums only ever move long-term in one direction: up.
More or less, I agree. With fewer accidents and fewer drivers insurance companies will be looking at decreased revenues. No company wants decreased revenues of that sort. Now, I'm not 100% familiar with the insurance industry and any regulatory capture that may take place, but then would be the time for a cartel to flex its muscles and ensure that startup insurance companies don't undercut them with fair pricing.
It would be cool if what you describe actually happened, but in the real world, premiums only ever move long-term in one direction: up.