Though it might come as a shock to many people. Very few doctors actually wanted to be the ones in charge of treating patients who were high profile.
Dr Stephen Arlington belonged to the vast majority of medical professionals who did not enjoy treating high level patients. Things like 'doctor to the stars' might have sounded good to say, and the pay was also astronomical. Especially if they decided to hire you privately.
But the negatives were just too much for most people. Not only would you have multiple people hovering over you. People with the ability to end your career with the snap of a finger if they wanted to.
The internal pressure was also something else. Medicine was stressful enough on a regular basis. It did not help matters knowing that the life of a well-loved celebrity, or of a president or an important diplomat depended on you.