If there's no leak, then inhibitor put into the header tank will not disperse through the system because there is no flow from header tank into the rest of the system.What would happen if I just put two bottles into the header tank without draining? Presumably it would eventually disperse through the system.
To get inhibitor into the system, you need to turn off the water supply to the header tank and drain off enough water to empty the header tank. Then add the inhibitor and turn the water to the header tank back on to refill the system and draw the inhibitor into the system.
Alternatively, close both valves on a radiator, drain it out, and add the inhibitor to the radiator by removing the bleed valve or the plug at the other end and using a screw-in tube and funnel (available via a search online)