They don't just say it, but they essentially have to. The way you negotiate with an insurance company is you threaten to go out of network if they don't meet your demands. When that happens, every one of their patients gets a big bill from you for your full charge and then they have to personally pay it and later seek reimbursement from their insurance company. It's hard on you since your time to collection goes up and your collection rate may go down, but it's terrible for the insurance company. Their patients revolt and they likely lose business as employers switch to other insurance companies. We went out of network once and they've never batted on eye on our demands since.
You think they will risk going out of network with an AMC that covers a huge number of cases? Zero chance.
It's all about negotiating and the more cases you do per year, the more leverage you have. AMCs have massive leverage.