Some patients do benefit from it, and not just the affluent patients.
A working class family can either pay $1k+ per month towards a HMO (and more if they want PPO) or they can pay the same amount (or in some cases, even less) towards a cash-only practice (whether it's a set fee-schedule, or a monthly fee or yearly fee, etc).
Remember, a healthy working class family with little healthcare utlitization is paying such a high monthly premium to help the insurance company pay for care of the infirm elderly (those that are polypharmacy and require multiple specialties and multiple hospitalizations), those in the ICU, etc. They also pay the insurance company to hire the many middle-people whose sole responsibility is to make sure there is "proper utlization" of healthcare resources (and also to pay the CEO and Board of Directors).
Instead, the same working class family can take their monthly premiums and apply it towards a concierge practice ... where doctors don't have to deal with paperwork and requirements of various insurance companies, a 5-15% rejection of payment rate, getting paid 6-9 months later for service rendered, etc. No need to hire medical claims specialists just to fill out forms to bill (at an already pre-negotiated discounted rate). The immediate savings can be passed along to their patients, to allow for a competitive rate.
Market forces at work. The insurance companies now must find ways to attractive healthy patients back into their fold (increase services or benefits, lower premiums, increase customer satisfaction, etc)
Addenum: The patient also benefit in other ways. Because the physician is no longer taking insurance patients, there is no longer that financial pressure to see a patient every 15 minutes to pay the bills, staff, malpractice insurance, and one's income. With lower overhead cost (and likely the physician will be paid more per patient that under the old insurance plan), the physician can take a longer time to talk to the patient, listen to their concerns, maybe counsel them, etc.