A drill press is a big drill mounted on a stand with a crank mounted on the side. When you turn the crank the drill presses down. A drill press operator would literally take something, place it under the drill, turn the crank to drill a hole in it, then remove the item.
I could see a big shop having enough work to support a full time drill operator but it is rather like saying you are a hammer operator or a screw driver operator. It isn't complicated enough to need a specialist.
"It isn't complicated enough to need a specialist."
Have you ever tried to center a hole to micron precision? Do you know which drill speed vs feed rate to use for the metal you are drilling? Do you know which drill bit to use? Do you know how to sharpen drill bits correctly? Do you know how to secure work to the table? Do you know how to use a sine bar and gauge blocks to mount your work piece at a precise angle? Do you know how to inspect the surface of the work piece to make sure it doesn't have burrs, specs of dirt etc that could throw the work tolerances out? etc etc
It's not he most in depth job, but it's far from unspecialised.