It's an egg-and-chicken problem then; you cannot manage the "available health resources" if those are nonexistent, so having those is a priority and that is more similar to what the BMGF is already doing.
A lot of countries in the developing world have existing health care resources, such as doctors or nurses, but managing them is a major problem in a state such as namibia or botswana where the population of a small city is spread out on a territory as large as germany. It helps if you can have a local, less qualified health care worker communicate with a remote specialist before either moving a doctor to the patient or the patient to the doctor. A medium to high bandwidth connection, eventually capable of video transmission helps a lot here.
You are painting a fairy tale outcome, poorly based on reality. Even in Colombia, were there is high-speed internet connections the coordination, willingness and education to make a fruitful use of this resource is scarce. So it's not like "Give them internet and they will figure it out!", it's a lot more complicated than that.
That's not what I'm saying. The same could be said about a malaria vaccine. "Develop it and all will be good" is a fairy tale outcome as well. Even "develop it, make it cheap and malaria will cease to be a problem" is a fairy tale outcome. The real world tends to be much more complicated than that - all I'm trying to point out is that there is a real problem managing available resources, especially in countries with low population densities. We can even see that in some less populated areas of eastern Germany (some areas of Mecklenburg-Vorpommern), but at least we in Germany can luckily afford to throw money at that problem.
We all agree is a problem, but we don't agree in that Internet is going to fix that just by existing. So in that order of ideas _by itself_ internet is far from being a priority over malaria and other set of issues.