Here in the US, you can get surprise medical bills greater than $10k USD, due to these organizations saying "nope, not gonna pay for that."
While in some cases, homeotherapy, and similar, this is justified ... in many/most cases, it really isn't. HMO/PPO etc are all about reducing cost of the care, with these organizations funding themselves from driving down the cost of the the service.
Doctors have to hire people to handle dealing with the insurance companies (HMO/PPO/etc.) Its expensive and time consuming to deal with them.
Edit: Kaiser is non profit. There are HMOs which exist primarily as cost control organizations, not primarily committed to health care, so experienced vary
HMO - health maintenance organization. It's a type of insurance where you have to get referrals from someone you designate as your primary care provider.
PPO - preferred provider organization. Another type of insurance where some providers are "preferred providers" who partner with the insurance company to offer reduced rates. You don't need a referral to see a specialist, but many specialists won't see you without a referral anyway.
I have no idea what a "PPO doctor" is, I've never heard of a doctor that only accepts PPOs, that's ridiculous. I guess if a referral is required then the nebulous "they" won't refer you to the "good" ones if you have an HMO?? Except, in practice, you end up at the same handful of providers in your area no matter your insurance. I've had both HMOs and PPOs throughout the years, same shitty doctors on both.
This is very dependent on where you live and the density of providers. If you live in a small-to-medium size city or smaller, your description is accurate. In big cities, it's not; there is a whole set of doctors who don't work with HMOs at all, because they don't have to. They often are the best, most in demand doctors as well.
“A terrible guide to the terrible terminology of U.S. Health Insurance” by Brian David Gilbert