I'll bite. I think, as with most industries, that private "control" of aviation, focused on profits, brings about pros and cons. While it's certainly true that insurance companies want to mitigate liability, and thus hypothetically want to do things "right" in order to avoid it, all one has to do is look back at the Ford Pinto fiasco, where the court was made aware of the fact that Ford knew the car has a significant defect with horrendous safety concerns, yet their legal department had figured that the risk of a huge lawsuit was low, so the accounting department decided that it was cheaper to take the risk (assumed to be small) of being sued over doing the "right" thing of recalling the cars (very expensive) and fixing the defect. This small detail was the impetus for the significant punitive damages awarded in the case, but, in my opinion, you're an idiot if you think similar thought practices don't go on today.
While I agree that in Pt 91 ops, which are largely untouched by the FAA, things are very safe, one must realize that it is not just the insurance companies ensuring so. The companies themselves recognize the caliber of employees that are put on those aircraft, and they are very concerned with pilot qualifications, maintenance standards, informal rest "rules," and the like. Going back to the Ford Pinto parallel, I'm not sure that the airlines would see things the same way (one just has to look at current management, especially at the regional level, to see this).