@sstainba,
I don't think that data is misleading. Most physicians are in private practice, either self-employed or employed by other physicians.
A doctor who works for a healthcare organization is still working in the private sector, and many of those are self-employed independent contractors.
Doctors who work in a clinic that is part of a hospital's outpatient services would be part of the 19% of physicians and surgeons employed by hospitals.
Most doctors work in some sort of private practice setting, including large group practices or private "clinics". Medical practice in the U.S. is very much a business. Doctors also own many private hospitals and nursing homes, outpatient radiology centers, sports medicine centers, cardiology testing centers, etc.