If you have a standard overflow tank and not a surge tank, it can have coolant in it if there is a leak in your coolant system. The simple reason is, the system works on a basis of pressure/suction. When the system is under pressure, if there's too much pressure, it will push a small amount of coolant out into the overflow bottle. Then, when the system cools down, it is under suction (or a vacuum), which will draw coolant back in from the overflow. It does this via the radiator cap, which is a 2-way valve. If you look at your overflow tank, the tubing coming from the radiator will be at the bottom of the overflow so as to always have the fluid readily available.
When there is a leak in the system, there is no way for the system to create suction (or pressure for that matter, either) as it can draw air directly from the leakage area. (NOTE: It can also draw combustion gasses from a leaking head gasket, which equates to the exact same thing ... it's just doing it a little differently.) Since the system cannot create suction through the normal means, whatever is in the overflow will remain there. There is no way for it to get back into the coolant system.
As for a surge type tank, it will show as empty, usually when cold. The reason for this is, there is a pressurized path (meaning coolant under pressure, which creates flow) from the water pump (or coolant system) which pushes fluid into the tank during normal operation as long as the pump has fluid to push. Once the engine is shut off, the air in the system will raise to the top, which should be the surge tank, which means all of the fluid will settle down into the bottom parts of the coolant system.
Bottom line is, yes, people should be checking their complete system for health, not just the overflow tank.