But Tylendel was chosen in the first place, wasn't he? Then gets a special job when his Companion dies iirc.

I suppose you could call whatsisname the engineer in that last series of Valdemar books an "Unchosen" but he has special skills that make him part of the elite group. I think I remember one short story either by Lackey or by a fanfic writer, in one of the Valdemar story collections, about a woman who does not get chosen, and it was actually one of the more powerful in the collection.