Its not really a sore throat, why would you treat it like that? If the dragonborn can't shout, he's not really a dragonborn anymore, if I was akatosh, I would obviously fix this for the fact that he can not fulfill his destiny if he could not shout. If you become a daedric prince in oblivion, thats not necessarily their choice, thats the daedra's choice. If the daedra allow someone to become a daedric prince, the gods can't do shit about that.
Bryn, it is reasonable to assume that using the thu'um would leave the Dragonborn weak, but the way of the Voice, or shouting is a form of magic. I assume that a novice user of the Voice would experience more physical side effects, but the more experience one gains, then hopefully those side effects would disappear. Additionally, master users of the Thu'um are rumored to be able to whisper because their voice is so concentrated with power (and I think this has already been mentioned).
As with any magic user, they need time to regenerate their ability to use magic. I think a follower of the Way of the Voice would also need recovery time, too.
I suppose someone could become mute if they over-extend themselves...knowing some of the methods but not completely experienced in the study. It might be similar to those who attempt to read Elder Scrolls and are struck blind immediately...they have enough knowledge to begin to read, but not enough to prevent the after effects of a glimpse into time itself. "A little knowledge is a dangerous thing..." and I emphasize the little.
Am I the only one who doesn't believe that "Shouts" actually use the throat at all? It seems to me that since they are a branch of magic may not actually use his larynx at all. I would assume that shouts are channeled through his voice/throat but that they can't actually "jack up" his voice.
Exactly think about it this way. For a regular person, it might just be like shouting at someone, but if you the ability of the thu'um it'll turn that shout in fire breathe or unrelenting force depending on the words you say. So when you shout maybe you're not shouting that loud, its just the magic from the words amplifying it like that. So maybe he doesn't actually have fire or frost coming out of his throat, but just a regular shout, and when the words or word is spoken, it transform into that, so it probably starts at the end of the mouth or your tonsils instead of your vocal chords, so you most likely wouldn't go mute in my opinion.