I have a level 3 doctor (competent) and I see 2 injuries/season in average that last one week or more (so it's not healed immediately after the game). That said, I rarely play Patient where one player receives a lot of fouls; but I like to play PtB and LI where my main PG (Zorin) drives a lot, and he's been injured only once for 2 weeks in 4 seasons.
So maybe it's not only doctors, but also aggresive opponents and tactics? Maybe bad defense from your opponents also makes them make more fouls on your players. This could be especially true in earlier rounds of cup where you encounter bots and D.IV teams.