The Doctor referred to himself as a "supplement" to the medical staff. In an emergency situation expertise and manpower would be at a premium. The TNG Tech Manual claims most Starfleet personnel are cross-trained in medical treatment for emergency situations. Certainly Worf was capable of delivering a baby in an emergency situation, so it may be true. But in a real crunch, an expert program would be superior to a cross-trained security guard. Any abilities outside of emergency medicine are unnecessary in an EMH. It only exists for "crunch time" situations. And if the primary physician is dead, that's a real emergency.
Dr. Zimmerman may have had a long-term medical holographic program in mind during the development of the EMH. If he did, any extra learning and growth potential in the EMH may have been included so that the LMH would be easier to develop - an upgrade of the EMH rather than a completely new project from scratch.