Heat going cold when idling

When you drive the car, at higher rpms the water pump flows better. So the high temperatures leave your engine slower at lower rpm (idle). I would try to get the engine hot, drive it and then stop the car but keep revving the engine. That way your pump will work like it was being driven.

If that way the heat doesnt go off I would assume you have some kind of flow problem into the heater core (as the car doesnt overheats)

Problems for flow into a heater core would be a dirt in the inside of the core, partially clogged hoses going into or out the heater or the pump itself. But if the rest of the car has normal temperature. I would say its unlikely the pump.

I know you said the line is already flushed. But maybe hasnt been done properly? If you flush your coolant system, you need to add some cleaner and run it for a while, at least until the heater warmps up.

The same thing that prevents hot water into the heater core, may prevent the cleaning agent to clean the core.