One way to look at it is that this Byzantine process helps Anthem by trapping valid claims in a black hole of bureaucracy. That way they don't need to pay, right?
It's not quite that simple. It's true that incentives are not usually aligned, but in reality the payment P&L impact is often far away from the claims processing administrative decisions. At many payers (like Anthem), the ultimate 'payer' for most of the cost may not actually be Anthem. They may be the third-party administrator charging back to an employer, or the Federal government. Even if it is Anthem, the owner of the cost center for this broken administrative process is almost certainly removed from whomever runs the actuarial accounting to decide what kinds of services to cover and how much to charge customers (except, as the author found, once they unite at the level of the CEO).
Ultimately for Anthem, this claim cost them a lot more to handle than if they had just paid out in the first place - even if they were the ultimate payer. Consider how much time the author spent on the phone with customer service representatives. Many of those were almost certainly contracted out and charged to Anthem by the minute. Then consider the claims processing backend, also likely contracted through third-party claims processing and management systems. It would have been simpler to manage the first one straight through.
Usually what it comes down to is that it is hard to estimate the ROI for investing in fixing a problem like this, because the benefits are spread out over many small cases. One way to prove it out is to highlight some of those small cases so that payers feel the pain and recognize the need for change. Thus, I am glad the author published this complete story.