There are cash-only doctors, or doctors who will accept cash as payment, and for what it's worth, the discounts for not using insurance can be substantial.

More importantly to me though was this less-talked-about perk of having a doctor who -- once he knows he's not just going to receive a negotiated rate from an insurer -- will actually take a little bit longer with you and hear out your complaints. Knowing that if you take longer, he can bill you more is actually a net good, IMO.