I was toying around with the concept of the undead a while back and I wanted to revisit it. Ultimately, I wanted to change the paradigm around. Rather than having the players being able to trivialize the undead with a Cleric in the party, I wanted that dynamic to work the other way.
Every undead creature has an innate ability to terrorize living beings and send them screaming away in horror or cower in fear. In the case of the Zombie, they exude a "fear of the grave" that requires a wisdom saving throw. So instead of every zombie encounter starting with a turn undead, it now starts with a "turn living."
As I said before, the undead should be something terrifying to characters and even a little bit for the players. With a turn living type ability and a re-imagining of their other abilities, undead encounters should be something special and a learning opportunity. We can create the opportunity for players to become undead hunters, rather than there being one in every church already.
So the corresponding change would have to come upon turn undead or else a battle with the undead just becomes a battle of saving throws. Let's change turn undead to "Faith" and it allows anyone who failed their "turn living" check to reroll. Of course, this supposes that the Cleric himself succeeded in that initial save.
Now, a Cleric becomes a weapon against the undead rather than an I Win Button.