I tend to view the Taint as a sentient being (of sorts), its not just an infection or simple virus; well I mean it does infect people but its more of a symbiotic organism than a disease. Sure you have infected Humans, Elves, Dwarves, and Qunari in the form of Ghouls, but Darkspawn born from a Broodmother aren't infected members of those races; Hurlocks, Genlocks, Shrieks, and Ogres are Darkspawn, a completely separate species. The Taint is deadly to most everything in Thedas its true, but its not a true parasite. A parasite lives off its host without providing any benefits, but the Darkspawn are functionally immortal, and they don't need to eat, or sleep; those are some pretty substantial benefits for a supposed killer plague to provide to its hosts. I think there is a level of nuance to the relationship beyond a simple disease.
The Taint/Darkspawn (IMO) are too complex to be a simple mistake or accidental byproduct of some spell gone awry or the result of some angered god. Sure there are clear examples of the Taint behaving akin to radiation poisoning or a virulent infection, but their are other instances where the Taint is shown to provide clear advantages to those it has infected. Not only that but the Darkspawn and the Taint are too interconnected to be written off as mere hosts of a disease; the Darkspawn Hive Mind, the Calling/Song of the old gods, etc. One can almost use the two terms interchangeably due to how dependent they are on each other.
I think that the Taint has always existed in the setting and it and the Darkspawn are a fifth intelligence species (Humans, Elves, Dwarves, and Qunari being the other four) living on Thedas.
EDIT: Another thing to consider is how the Taint not the Fade is the source of the Darkspawns' magic, and how the Darspawn have no souls (i.e. presence within the fade, and not just muted like the Dwarves). That (to me) is far too complex to be some hiccup of a spell, I mean the Taint is the only thing that can draw on something other than the Fade to create magic.