Before 1850, dentists were not yet classed as doctors and this also applied, believe it or not, to surgeons. They were classed as tradespeople and were treated as such by the medical profession. This attitude continued for some time and was still ongoing during the civil war although attitudes were changing. One of the reasons that barbers poles are red and white is that barbers,were often barber/surgeons too.
Surgeon = the guy who did the messy, 'hands on' work, applying and changing dressings, lancing boils, setting bones, blood-letting, etc.
An aside.
The origin of the barber's pole appears to be associated with his service of bloodletting. The original pole has a brass basin at its top representing the vessel in which leeches were kept and also represents the basin that received the blood. The pole itself represented the staff that the patient held onto during the operation. The red and white stripes represented the bandages used during the procedure, red for the bandages stained with blood during the operation and white for the clean bandages. The bandages would be hung out to dry after washing on the pole and would blow and twist together forming the spiral pattern similar to the modern-day barber pole. (YES the bandages were re-used after washing. Hygiene only went so far.)
View attachment 592648
The Americans added a blue stripe as a nod to the national colors.
In England, the barbers were chartered as a guild by Edward IV in 1462 as the Company of Barbers. The surgeons formed a guild 30 years later and the two companies were united by the statute of Henry VIII in 1540 under the name of the United Barber Surgeon's Company. In actual practice, however, barbers who cut hair and gave shaves were forbidden to practice surgery except for bloodletting and pulling teeth and surgeons were prohibited from "the Barbery of shaving."
From:
https://www.pallmallbarbers.com/mens-grooming/history-barbers-pole/