The red light only glows when the fuse is removed (or blown) if there is a load on the circuit. If the ceiling light is off or not working due to a broken connection, and nothing else is on that circuit, the LED will not light up. It relies on the circuit downstream for a current return. The circuit is quite simple: just an LED and a suitable dropping resistor in parallel with the fuse holder.
If all the switches are on (including those on the lamp), it sure sounds like an open connection somewhere. I'd start by looking at the light fixture connections and the switch connections, and testing the switch itself (by e.g. temporarily shorting it out).