Their role in the system is to more or less to do rationing / make resource allocation decisions (or at least be the "administrative face" of rationing policies) and this is always going to be unpopular. While "look inward" is always good advice, I think these companies understand that some percentage of customers will hate them simply because that is the nature of the business.
They have to say "no" to some things. The more things they say "yes" to the higher your premiums are. If they say yes to way too many things (like public health care systems will tend to do) then you will see rationing by waitlist rather than price. They are also subject to the laws of large companies and so will make both random and systemic mistakes (just like every other large organisation) for the usual bureaucratic reasons.
Even if major systemic change is made, whatever entity is left with the responsibility of saying "no, you don't get that care" will still be hated.