Jason has provided a good short answer. Expanding this a bit, logistics is an important often overlooked element. Part one is the distribution and scheduling of key personnel (more so when under-staffed). Part two is getting an adequate supply of appropriate material and drugs to stores before they run out. Part three is adequate maintenance of of equipment, buildings and the supply chain.Planning for these 3 parts requires capacity to a) see beyond the immediate crisis and take account of future need (forecasting information) as well as lags through the bureaucracy (hospital to national medical agency) and supply delivery time requirements and b) provide forward funding. Many advanced economies find this difficult, especially their politicians, so it is not too surprising that less developed economies, with usually even fewer far-sighted, public minded officials to hand, have major problems delivering any sort of health care.
In my opinion education and motivation are the main issues. Second is the hardly controllable power of the farm industry. Third, I subscribe to your views.
I recommend the empirical work of Dr. Bruno Yawe who has used non stochastic frontier production functions to study the efficiency of hospitals in Uganda and other African countries. Some of his work is available online. Skm