A few relevant items from someone working in the industry.
1. Auto insurance is regulated. The insurer has to file rates with the state, and they cannot deviate from those rates. If they offer to lower your rate, one of the following is happening: They gave you a discount you didn't previously have, removed coverage or reduced limits or increased deductibles, or they have corrected some information that was previously not correct.
Before switching, things you can do is ask the company to go over all available discounts and determine if you are eligible for something you haven't been getting, discuss options for reducing coverage, and confirm that your consumer report information is correct (this includes your CLUE report, MVR and credit report.)
2. Most companies will request at least name, BD, SSN, address, email &/or phone. Giving you a quote costs them money, from $.50 to $20 depending on your state and the accuracy desired in the quote. Most cost around $2 per person on the policy. They don't want to spend the money unless they are reasonably certain there is a chance of a sale. If you don't provide contact info, your odds of conversion are tiny. Why would they blow $2?
3. For pure online quotes, esurance, geico, progressive, 21st century are reasonable companies. If you are getting put into the queue for a phone call from these companies its because you don't fit their underwriting criteria so they slow-track you, which they know means you will likely buy elsewhere. You might also try Root. They have you download an app and if you are good enough, they will give you a quote after monitoring your driving for a while.
4. I've no love for Amfam, but their claims reputation isn't bad. State Farm also has a solid reputation for being easy to work with. Progressive, Geico and Allstate are all known for fighting claims tooth and nail. That said, I'd have no reservations about being insured by any of these companies, as all are financially sound and do pay claims when coverage is identified. I just know if I go for the low-cost direct company that my claim will be slower and more frustrating.
5. Each carrier makes up its own rates, so odds are that you will find a wide range of available prices. That said, make sure you understand each quote. Are the coverages deductibles and limits the same? Are you paying for expensive options like accident forgiveness (or prepaid accident surcharge as the actuaries call it) or new car replacement?
