With the horrible reputation US insurance gets, is there some reason why smaller insurance companies aren't popping up to deliver better quality (i.e. less denials)? Is the field simply heavily protected by lobbying and regulatory capture?
My faith in things is so entirely shot that if I saw a company like the one you suggested, my immediate, heels-dug-in assumption would be, "That's cute, and in five years they'll be just like the rest of them".
Due to a weird, historic tax incentive, people get mostly get insurance through their employers. Their interests are not in opposition to, but are still different than those of the people actually being insured. So you have a situation where the people doing the insurance "shopping" aren't the ones using the insurance.
Then on the actual healthcare side, for no coherent reason, almost all prices are hidden from the patient, totally inscrutable, or highly variable. Providers and insurers collude to keep these prices astronomically high so the providers they can overcharge medicare (which is not allowed to negotiate prices) and the insurers can "give" their customers "negotiated prices."
A more normal free market would be something like: people buy their own insurance or pay providers out of pocket. This would incentivize people to make price conscious decisions and evaluate insurers and providers for quality. Providers would need be required to set and publish fixed prices for different procedures and services so the market could actually function, but this seems pretty simple.
These days the bedside manner is trash. Though I don't blame the doctors. And they are little more than privileged employees...if that. The real service they provide, and the service people are growing to expect, is to navigate the insurance system to wrangle out some semblance of coverage. Being employees the most you are going to get is a wink and a nod because as employees they cannot speak freely.
In the case of healthcare I don't think we are capable of building a system this large, prescribing every detail, while getting it right. It's like trying to plan an economy.
I am no expert, but my impression is that it would be very hard for the insurer's customers (e.g. employers) to know an insurer was better, and many might not care. There's so much opaqueness and indirection in the system, it's not surprising it's both inefficient and heartless.
That said, all systems ration in some way or another. We need to come to grips with that.
If you happen to have a working spouse, you can choose between which work coverage makes most sense for your family.
This also means that any new insurance company needs to pick up a ton of customers. This is really hard to get off the ground in a way that will make you money. You would ideally pick up a healthy population (younger folks), and then you'd have to get lucky on your actuarial tables for the population of people you pick up insurance plans for.