I quite like this approach, as it happens. My all-time favorite undead are the Cauldron-Born from Lloyd Alexander's "Chronicles of Prydain," whose signature trait was that you could not kill them. Ever. By any means. There was no victory against the Cauldron-Born. The best you could do was delay them until they were forced to return to their places of power, since they weakened the longer they were away.
I would really love to see something like the Cauldron-Born in D&D. However, I would not consider this treatment appropriate for skeletons, which are well established in D&D's history as the fragile, disposable fodder of undead armies. Better to create a new, more formidable class of undead, and then just say skeletons don't exist in that setting.
The closest I ever saw to something almost as evocative as the Cauldron-Born were the Fell from Fantasy Flight's
Midnight campaign setting. In that world, the land itself is so corrupt that there is a good chance that the recently dead will rise - with a burning hunger for the flesh of the living.
A newly risen Fell is very similar to what it was in life, but must feed on the flesh of sentients periodically in order to keep from decaying. Fell of this sort are quite cunning, and even capable of passing for living - if they're careful to hide their injuries. However, if one of these creatures goes long enough between feedings, it degrades. The lesser Fell are basically slightly tougher versions of ghouls, zombies, and skeletons. Once they degrade, feeding keeps them at that level, but they can't ever return to the higher state. And even though they can no longer feed, the skeleton-type Fell have nothing left but the need to kill. And none of the heroes in the world of
Midnight have the ability to turn undead, which makes the hordes of risen dead even scarier.
Unlike most undead in D&D, the Fell are downright f'ing creepy. They're an important element of the setting which helps to set the tone in
Midnight. But that's really the only time I've seen the undead used in D&D as anything other than a mostly boring low-level threat. Which by all rights the undead shouldn't be, but are.
In standard D&D, you basically need wraiths to scare people.