How long could a pilot/co-pilot of an airliner doze off?

No, because there is no way for ATC to transmit instructions to the auto-pilot. ATC can not remote control any airplane, and given that most air traffic controllers are not pilots you probably wouldn’t want them to be able to do that, anyhow.

“Back in the day” there was a time when you almost had to take a nap to survive. Without an autopilot it becomes very tedious to maintain course, altitude and speed. One person would fly for up to an hour and then switch off. It is a mentally draining task to maintain an IFR flight plan. You have to constantly scan the gauges to make small corrections.