Here is how I think it works, someone correct me if I'm wrong
Injuries happen during the game, on fouls. There is two types of injuries:
1. Mild injuries - Your player cannot continue the game, but is healthy after the game. If your doctor has taping injuries specialty, the player can return during the match. I'm not sure if players can return during the match with ordinary doctors.
2. Heavy injuries - Your player cannot continue the game, and is injured for X amount of weeks, after the game. The better your doctor is, the less the amount of weeks is.
That means, your doctor level matters only during the game in which the injury happened. It's pointless to buy a better doctor after the injury. He doesn't "heal" the player, he only sets the initial, after the game, amount of weeks.
I'm not sure on this: Are the mild/heavy injuries set, or they depend on the level of doctor. Let's say a player was injured and the doctor was level atrocious. He gets a 2 weeks injury. If the doctor was better, would there be a chance, that player is healthy after the game?
What you said is what I had been told was true.
However, after reading the other posts in this thread, perhaps it is not the case.
Sorry for any misinformation.