Because they exist only because of the corruption.
That's what's bugging me. The taint is a force that brings death in its wake. It's the antithesis of life. How can it be a force that sustains life, even decaying lifeforms such as the darkspawn? Have you ever heard of a disease that keeps anyone alive?
Which is why I think the darkspawn are not kept alive because of the taint. My belief is that this is a misconception which we, the players, were forced into by lack of clear evidence in the games as to what the taint truly is.
Whatever keeps them alive may not be the taint. It may be something connected to the ancient Magisters that first entered the Fade - the first darkspawn in existence - and their unexplained connection to dragons (Archdemons). I personally believe it might have something to do with the secret of effective immortality (Solas in Trespasser), the blood of Titans (lyrium) and sharing your soul with other living creatures such as dragons, considered to be powerful in the DA lore.
The taint might just be the darkness found in the Black City, nothing more. Some form of twisted magic from Arlathan and the Evanuris that infected the Magisters. What keeps darkspawn alive may be something entirely different. Some form of blood magic practised by the Magisters, tied to the blood of dragons, perhaps. Their blood is supposed to be special, since they were revered creatures, mistaken as gods by old Tevinter.
The taint might not be the key to understanding the darkspawn immortality at all. That's what I'm trying to figure out.