Actually Hexing is nothing like EMP, it just looks that way at first glance since modern digital is the first thing to go out. Hexing can stop a steam train if the wizard is old enough, that's not something EMP can do.
As for the government or military or any larger organized having a supernatural branch, I don't see it happening. Not on a large scale in any case. See say a small part of the FBI is supernaturally aware and uses it in in their work to gain an edge over the "bad guys" seems like an interesting campaign actually. But should they grow, they will need to acquire space and people and money and material. Sooner or later they will have some sub division or whatever, it will have a name a stated purpose and quarterly numbers of what they did. Now perhaps you can get a guy one or two rugs up the ladder to help your division out for a while and keep it on a need to know basis. But sooner or later someone up the line with more stars or whatever is going to see that name in a report, see the resources it is eating and what they are supposed to do and he is going to shut it or down or seriously decrease their budget at the very least. Its a economic crisis out there and we don't have the funds to frivolously spend on whatever your doing down there. No I wont come and look at what it is, I'm a rational man who doesn't believe in magic and besides no good Christian would allow such things, god forbid.
The way governments are run don't lend themselves well to magic focused departments, it simply doesn't fly for long. As mentioned tough a division focused on information, especially history and the like will be far more effective at breeding a in the know crowd. However these sort of organizations are less do and more observe and anallize in nature most of the time.
Mind all of that is purely my 2 cents.
