Perverse incentive structures are the fundamental reason US healthcare is so expensive and stupid. It's not "greed" in a simple sense, and it's not "regulation" in a simple sense. It's a system of interacting problems that needs treated in a systemic way.
Blaming the insurance companies strikes me as the political equivalent of yelling at the cashier.
Insurance is one of the oldest industries in the US, which means they have accumulated many (probably well meaning, but ultimately stupid) regulations over the years.
For example, to be taxed as an insurance company (e.g. avoid paying tax on premiums that will partially be paid out in the future), most states assign a maximum profit margin.
That means, in order to make more money as an insurance company, it's perversely in their best interests to increase the cost of medical care as much as possible. Medical care is not like cat bonds that might wipe out your risk pool, so it's just a cashflow game.
That's not to say there isn't any downward pressure. On a day to day basis, individual claimants eat more into that margin than was budgeted, so a company might have to raise rates higher than their competitors (essentially they have some pressure to deny claims), but there is very little systematic pressure to reduce overall costs. Also, marketing budget does not count towards profit, which is why insurance companies are in this rat race of brand advertising to steal customers from each other. With auto insurance, for example, upwards of 40% of your premium is just going out the door as ad spend.
Anyway, I would assign a lot of blame in this perverse incentive scheme to poor regulations.
To put it in software terms, it's not a bug, it's an architecture flaw. The "poor regulations" aren't individual issues, but rather part of the fabric of a framework that causes perverse incentives. Therefore, the simpleminded conservative/rightwing solution to "reduce regulation" doesn't eliminate the perverse incentives - it just changes the flow, the way tossing a rock in a river does.
If I were the one making the decision, I'd change American healthcare with one big thing, modeled on the Japanese system... price controls. Japan's system is more or less like ours, but with one key difference - a government panel sets the prices for all medical goods and services. Providers and insurers have to make their profits within that price structure. This incentivizes both to reduce costs, for the sake of their own profits.
Of course, this would be called radical socialism or some such stupid in America, and Democrats aren't even talking about it (Republicans have no ideas at all on health care). But it's politically a much lower threshold than single-payer (throwing out the whole framework and starting over, basically). And it demonstrably works - Japan's system costs less than half what ours does per capita, with better outcomes.
Blaming the insurance companies strikes me as the political equivalent of yelling at the cashier.