The EMH seems to think that his hologrammatic status means he's better than everyone else. Well, to blazes with that. He is still a sentient being, and thus has the same decision making processes as every member of the crew.
Just because he's a hologram doesn't mean he makes better decisions or fewer mistakes.
Doctors with that much experience no longer are affected by the deaths of their patients. Well, they are but only temporarily and they can learn of the death of a patient in the middle of the afternoon and still have a good time in the evening. They wouldn't be able to enjoy life otherwise.