Oversight is not the only duty of shepherding, but it is the one Peter mentions here in the situation of suffering. Elder-shepherds exercise oversight. They lookout over the flock. God holds them accountable for seeing the big picture and acting for the good of the whole flock. But by virtue of their calling and their gifts and their affirmation by the church, they have a responsibility that is different than the rest of the sheep.
Responsibility is the key word, or accountability.
The rest of the sheep will not be called to give an account for oversight. Only the elders, the shepherds. Now as far as I can tell, all of that was universal in the early church. It didn't vary from church to church. It is not unique to a suffering congregation.