i disagree about calling auto shops and asking who gives you the least hassle - as long as your going with a large insurer just pick the lowest cost for the same coverage. basing insurance based on the service provided not cost doesnt really make a lot of sense IMO. unless costs are equal then considering the service should come into play.
I think it doesn't take the whole situation into account to shop by price only. If insurers A, B and C have reputations for faster payments, more generous payouts and less hassle than insurers D, E and F, then insurance from A, B and C have additional benefits to the deductible and insurance coverage and levels, presuming those are equal.
How much those additional, softer, benefits are worth to you is a different question.
I had some wind damage on a rental of mine last month; the insurance was with State Farm and the adjuster actively looked for ways to cover items that I would not have presumed would be covered then cut me a check on the spot. They definitely possessed those soft skills that I referenced above.
As a counterexample, a family member had a family of skunks take up residence in the HVAC duct work of one of their rentals. The pest company tech scared them and they sprayed...you can imagine the results. My family member had a hell of a time getting their cheapest possible insurer to pay out one red cent. They ended up paying for the help of a friend of a friend who worked in the insurance industry to read through their policy and help them fight the insurance company to pay for an unusual, but covered, situation. They eventually received a scant payout after several months.
How much is that worth? 50% higher premiums than the cheapest competitor? Probably not. 10%? For me, hell yeah.