Many Specialists focus on different things or believe different hypothesis. There is quite a bit of over lap in what they feel is going wrong in ME/CFS.
From what I have read which is a ton. Depending on who you talk to you may get a different response.
Peterson, Klimas, KDM, Dr. Montoya, Dr. Goldstein(no longer practicing), Dr. Lerner (past away) in my opinion have the most experience with regard to the amount of patients seen.
KDM, Klimas, Petterson and Lerner have all seen or treated over 10,000 patients. That is why they were chosen to develop the CCC Criteria.
KDM has seen 18,000 plus.
I personally would consider experience over everything in my opinion.