It isn't fully autonomous yet. For any future system sold as level 5 (or level 4?), I agree with your contention -- the manufacturer of the level 5 autonomous system is the one who bears primary liability and therefore should insure. "FSD" isn't even level 3.
(Though, there is still an element of owner/operator maintenance for level 4/5 vehicles -- e.g., if the owner fails to replace tires below 4/32", continues to operate the vehicle, and it causes an injury, that is partially the owner/operator's fault.)
Wouldn't that requirement completely kill any chance of a L5 system being profitable? If company X is making tons of self-driving cars, and now has to pay insurance for every single one, that's a mountain of cash. They'd go broke immediately.
I realize it would suck to be blamed for something the car did when you weren't driving it, but I'm not sure how else it could be financially feasible.