"That incorporated doctor provides medical service to a patient, not to a company that pays him."

The patient isn't trying to include that cost in their business activity. The company paying the doctor has the business relationship, and that is why they fall under the 1099 reporting rules. Even buying into your perspective, the 1099-Misc instructions include reporting payments you are making on behalf of another party, such as a law firm has a lawsuit which involves hiring a medical professional to do a patient assessment for purposes of negotiating a settlement amount. If you ever work with an employer who falls under DOT rules, they hire doctors and phlebotomists all the time, and they do lots of this reporting.

"Therefore the incorporated doctor does not need 1099 MISC. Am I correct here?"

Corporation reporting for medical is one of the many exceptions which require reporting. From the IRS 1099-Misc and -NEC: https://www.irs.gov/pub/irs-pdf/i1099mec.pdf