I officiate HS, College and Semi-pro football. All levels are kept on the field (though HS is a different official and not very accurate)
OK... I finally found it for the NFL
Game clock operators are hired by the league... but they live in the local geographic area of each stadium. It is the side judge's responsibility to keep an eye on the operation during the game. Replay CANNOT be used to check the clock operation (if that's the only thing in question) UNLESS it's the last play of either half.
What could go wrong with that. LOL
That's why there are still problems with it. Remember the Pittsburgh - San Diego clock fiasco a season or 2 ago when they let like 20 seconds run off on purpose. The league apologized for it after the fact... their response was basically that the side judge just didn't catch the mistake of the clock operator.
So yeah... they may be hired by the league. But they are not NFL officials and they live where the stadium is located. BTW... in the NFL... the stadium clock IS the official time.