I don't have experience as a physician or patient with concierge medicine. But as a mostly-retired specialist physician, I can see the idea of a concierge specialist. Many Primary care providers refuse to manage chronic conditions for their patients, even if those conditions are actually very, very common in their patient populations (i.e., they should have, and probably did learn all they need to know about managing those conditions in their training). For example, (not my fields, but just an example), high blood pressure, or managing cholesterol, etc. So, then patients HAVE to see a specialist and continue seeing that specialist for years and years with lots of on-going refills needed, chats by portal messages, acute phone calls when sick, etc. I had thousands of patients who I followed over many years who did not really ever need to see me, even for their first visit, let alone their 12th or 20th or 50th. Any willing, and basically competent PCP could have taken care of them. So, if the PCPs (and the patients to some extent -- some just want to be managed by a specialist no matter what) force this patient volume on specialists, then I can totally understand the idea of some specialists adopting the concierge model.
What I do now after retiring last year is working 4 days a month seeing only NEW patients and then I pass them to other specialists in our group for follow up. MOST of them do not need to be seen by a specialist at all because their problem is basic and any PCP should be able to manage it, but the PCPs do not even try to help these patients. The issue is so disturbing to me that it is a major source of dissatisfaction with my work, and even though I don't otherwise mind the 4 days of work and 26 days of "vacation" each month, I am going to quit this part time thing soon. I feel like I am helping perpetuate a stupid system of duplication, high costs, unnecessary layers of care, etc. I feel like being part of something that is stupid makes me stupid. The only thing that keeps me going is that I get a free Wall Street Journal subscription out of this deal because the university associated with my work has a library subscription that is extended for free to all of the "professors".