Been looking at every vehicle I have repaired over the last week (Sad ) and 99% feed water to the bottom hose from the bottom of the header tank and the top hose goes from rad to engine,some have bleed from top of rad to top of tank and others had feed from top hose to top of tank.

Logically thinking if you only feed water to one hose only where is it going to go,to achieve cooling you need to move the water through the rad to get heat transfer and therefore cooling, still convinced the two pipes coming from the filler neck in the picture are the answer to your problem and tying in those two heater pipes to the two pipes at the neck will give you flow.

Pm me if you want a phone number to discuss it.