I do apologize if you consider anything I wrote an insult; it certainly wasn't meant that way... just merely suggesting that things can be - and are - different based on where you're located. What works in Chicago doesn't necessarily apply everywhere. That's all
@Big Dog was getting at, and all I was trying to get you to consider.
There's several components to insurance - collision, comprehensive, and injury. No-fault doesn't mean a thing with collision or comprehensive, but it means
everything with injury. In a no-fault state,
no matter who is at fault, whenever there's any injury claim, YOUR insurance is the one responsible for handling the claim. Not the third party.
Even if it's as simple as being checked out by the ambulance squad; that would be covered under YOUR personal injury policy, not the third party's. Doesn't matter if they were drunk, ran into your car through a red light and hit you head on. It's still filed against your own policy, not the third party.
This is why I said it's best to talk to your agent. If there's any injury component whatsoever (and in most cases where a vehicle is totaled, there will be at least some injury component) - you'd have no choice but to file a claim with your own insurance company anyway. So there's no real downside to filing the entire claim via your own company - let their attorneys work for you ...
edit: Here's a great link that explains how it works:
https://www.iii.org/article/background-on-no-fault-auto-insurance