For the same reason that the game world is filled with magical healers, and not surgeons. One of the fascinating aspects of TES is that magic is treated as a mundane, commonplace thing (I really hope that Keyes' novel taps into this). In the TES world, scientific endeavour seems pointless - the Gods are real, and the powers of magic far outstrip anything that scientists can hope to currently achieve.
So why is magic/supernatural intervention needed to cure vampirism? Because scientific progress is slow if even nonexistent, and most likely always will be.
You are confusing many points here. Diseases may have magickal roots, but they are just that - diseases. Can they be cured by magickal means? Of course. However, they can also be cured by conventional means that do exist within Tamriel. Do game mechanics present this? No, just the same as game mechanics don't present linen bandages or wound cauterization. Should we argue, then, that since game mechanics don't present these things, they don't exist? http://search.freefind.com/find.html?oq=cauterize&id=72791987&pageid=r&_charset_=UTF-8&bcd=%C3%B7&scs=1&query=bandage&Find=Search&mode=ALL&search=all That would be silly. Therefore, traditional methodology of curing diseases (i.e. prepping oneself as best as possible for the immune system to fight it off, eating herbal concoctions to ease the symptoms or the source, etc) still exists, because in lore (as opposed to game-mechanics), mages are much rarer, and hardly anyone has the money to pay healers. And yet they aren't all shriveling up and dying en masse of the level to be expected from no other alternatives other than magickal intervention. Therefore, they must be using non-magickal means to treat their victims.
The point to note here is not that all diseases can be cured by magic and therefore, by virtue of being cured by magickal means, vampirism and lycanthropy mix in with diseases. No.
The point(s) to note are as follows:
1. Though diseases may be cured by magickal restorative healing spells/potions, those traditional restorative methods simply do not work on vampirism and lycanthropy. Because of this, the only solutions (very rare solutions, I might add) to vampirism and lycanthropy involve Daedric princes (specifically, Hircine, Molag Bal, and Vaermina) undoing said curses via their immense power, or the Witches of Glenmoril (who are often associated with Daedra trafficking and the like, are ridiculously powerful themselves, and therefore are influenced by Oblivion and are beyond the scope of "normal magic") mixing up a rare concoction to undo said curses. Therefore, vampirism and lycanthropy are outside the range of traditional "diseases."
2. More importantly, what you don't seem to be grasping overall, is that lycanthropy and vampirism aren't just diseases that popped out of unknown origins, like the rest of the random afflictions in Tamriel. Lycanthropy was created and set upon the world by Hircine; it is HIS curse/blessing. Vampirism was created and set upon the world by either Molag Bal or Vaermina, or a combination of both, depending on which source you cite; it is THEIR curse/blessing. The only reason that the Glenmoril Witches are even capable of curing such things is because they themselves are way more versed than even powerful mages about such Daedric curses, and they can mimic Hircine's/Molag Bal's/Vaermina's methods of undoing said curse/blessing.
Now, to get back to the origin of this, since these afflictions arise from Daedric sources, not mundane sources, the odds of mixing and matching are ridiculously low, because Hircine is not going to want to tie someone down to his sphere who is already tied to Vaermina's/Molag Bal's sphere, and Vaermina/Molag Bal is not going to want to tie someone down to his/her sphere who is already tied to Hircine's sphere.