And I was fortunate enough on any accident the other insurance company handled everything well. (One time I was lucky that I have Allstate, and so did the person who hit me.)
One accident my vehicle was totalled. I was told by their insurance company that once the air bags have been deployed, and the seat belts are locked up (from the wreck), they automatically total the vehicle. Which was in my case, and my vehicle was not quite 4 years old then.
I also asked my agent if my rates would go up if I filed through them and not the other insurance company. He said basically what your said.
Of course the rates have gone up anyway.

And although I have had Allstate for decades, and never had issues with them, I am insurance shopping anyway.
The ironic thing with my last wreck (that totaled my vehicle) Allstate agent said I had to file with the other insurance company for them to pay.
Okay, I did.
Then that insurance company, in addition to their claim, had me file one with Allstate.

I think it was all basically paperwork. But I found it ironic the other insurance company would not handle the claim I made with them without me filing through Allstate too.