You're totally right, and I'm not sure why this is so far down thread. The simple thing Economist could do is find countries with more doctors and talk about the natural experiment - you know, they could report on economics - but healthcare is bad "there," so maybe it doesn't fit their narrative.
To me, it's interesting that doctors in America are so well compensated seems associated with a large, well trained, and bureaucratically mature midlevel & support staff. There are synergies that unlock more complex care for sicker patients. Procedures done with that many people will be more expensive, the MD takes a smaller cut of a disproportionately larger pie.
For example, an operating room in a US academic hospital might have multiple anesthesiologists. In an African country with a lot of talent and many doctors nonetheless has zero. One take I've heard is there are too many doctors there, just like you are saying, except I heard this from an MD.
In my country the government flooded the market with doctors and the results weren't pretty to say the least.