In my limited experience with car accidents, they generally do.

This car burst into flames overnight while parked. Turns out it was an electrical short in the dashboard wiring. But, cars don't often self-immolate, so the suspicion of wrong-doing caused a delay.

I also don't know if they paid before the investigation was completely closed. They may have, but even at that, it was a few weeks.

They don’t do this out of the goodness of their hearts, insurance is ‘better to have and not need than need and not have’ and the longer you can go paying premiums without needing to file a claim the better for the insurer (monetarily speaking)