I know the gas-fired water heater in our house will not fire up without the water flowing. We can turn the water main off when we leave for a week or two and the water heater will not keep trying to heat the water. I suspect your system is the same.....? No water flowing = no fire.
Two totally different things, or systems. Your gas water heater is heating your potable water for your home and yes water flow must be there.
Water for heating such as baseboard heat, (radiators) as the OP said, is not part of the potable water system.
Water flow provided by an inline pump that supplies all the radiators with hot water is required to fire the heater, but no water flow from the line is required unless a low water alarm goes off and makeup water is required.
There are several safety checks on the system.
1. No water flowing through the system will cause the heater to build up steam by heating the same water too long. So a pump pushing water through the system must work.
2. There is a high temp or overheat sensor that will shut down the gas valve if it over heats.
3. There is a low water shut off that will shuts the boiler down if a low water situation is detected.
4. There is a pressure relief valve that will relieve the pressure created by any steam from overheating.
5. There is an expansion tank that has a air bladder in it to absorb the expatiation and contraction of the heated and cooled water.
6. There is an anti corrosion additive and antifreeze added to keep the pumps and lines free of corrosion, and from freezing and bursting if system shuts down in cold weather.
Because of this being a "closed" system, makeup water would only be required if water was leaking out somewhere within the system.
Our system here works with the NO water flow from the makeup line for years because there is no leaks.
So if the system is not working after water is shut off, it is low on water due to a leak, or one of the other safety system systems could be activated.