Your doctor's time is likely more valuable than your own.
In fact, their skills as a specialist likely outweigh the value of any of their patients.
Therefore, having to wait at your doctor's office makes sense.
You tell the patient to come at 9, but you don't plan on seeing them until 9:15. This gives them an acceptable cushion in case they're late, without making them too angry, and ensures the doctor's time is always being maximized.
What I don't understand is a perpetual hour-long wait. I don't understand the economics there, and yet I experience this all the time. And now, with a recent mass addition of healthcare participants to the game, things might get crazy-town.
Solve either problem.