My answer to the insurance question is always "it depends on which bin you are in each company".
Say you have some no point accident, many insurance companies would look at it as "no traffic violation", and then there are those with "we forgive 1 accident at fault" and there are those who "we count everything even if there is no point".
I have consistently gotten cheaper insurance quote from AAA than GEICO because somehow GEICO think I am higher risk than AAA. I have also seen my coworker, who came from S Korea with 10 years of driving record there, getting cheaper insurance from Progressive than GEICO, because GEICO doesn't include proven foreign driving record and treat him like a new driver.
So, about why your insurance is cheaper in one company than the other, it is not always one way or another. I have gotten cheaper home and auto split up for a long time because I cannot find one company that is cheaper to me in both together, they either bin my home as high risk or me as high risk driver, or they suck in either home or auto insurance.
Same for mortgage as well. Rocket Mortgage is always the worse for me because they have some rules and then their sales don't believe it when I found something that beat them by 0.5% because some wholesale deal a broker had with some bank needing a certain type of portfolio to bundle into MBS and they pay some of their commission into a cashback to me.
About service quality in claims: I had about 4 not at fault accidents in my area recently, and I typically ask the body shop on advices like which company is easy to deal with and not easy to deal with. My opinion is my own only based on the annoyance level at the repair, and so far GEICO and Progressive are the most annoying in the repair department and AAA is the easiest, with most others in between. All State want you to pay for your own rental car first and then they reimburse you later, so if you are cash flow negative then that may be something to be aware of.
I am glad I didn't have to deal with each company's lawyers yet, knock on wood. Another reason I took AAA is they have the highest amount of umbrella I can buy for my profile. Most of those others end up with a much lower number like 500k or 1M and AAA I think can go up by another 500k to 1M of coverage. In California you got to be careful of these things.