HI WB, It really doesn’t matter all that much which side of the radiator the valve is on (supply or return) with a non electric thermostatically controlled valve. ON HOT water Systems.. Steam is another story
Once the set point temperature is reached the valve closes and stops Or slows ( throttling condition) the flow for the desired temperature to be maintained and it doesnt matter if it is supply /return as the flow is stopped/ restricted.
Now if only half your radiator is hot and the other side is cold you have an air bound conditions which means the air must be bleed out as you cannot have air and water in the same place.
Most older H/W isolation valves cannot be a positive shut off as they are designed to have a constant circulation with a small orifice drilled in the valves cylinder or gate depending on the type you have.
The reason is to protect the system by have the water circulating to prevent freeze ups by forgetting to turn it on again.
In NYC it is not uncommon to find several variations of heating valves on the same system and most unfortunately do not say open or closed.
Some of the older ones even have a the same kind of throttling as an old gas cock BUT on hydronics they call them balancing valves as when they are installed they normally gave a gauge near by so each zone can be balanced by the return temperature GREAT SYSTEM but very old as nothing can go wrong once adjusted.
I think your best bet is to call a Licensed plumbing contractor and have them show you about the proper bleeding procedures. Good luck