I understand.
Many of these units use a "wheel" for the machine to sense water flow. If the wheel does not spin, the unit will not turn on. Further the element could be in need of cleaning. The wheel not spinning will not reduce flow. If the sensor that tells the heater that water is moving is not working properly, the unit will not fire. This is the same on ALL tankless heaters. It is the starter switch so to speak. That and a safety to protect the heater.
If it is actually drawing proper power, it would be heating at least some. Just because it is on, does not mean it is heating.
Flushing is recommended once per year on any tankless heater by in large.
Some small tankless units are not really made to be serviceable which is to say that it would be unlikely that you would find a plumber who is willing to work on it as opposed to replace it. I would not, just for liability purposes.
Look on Youtube for a tankless flushing video and give it a shot.
BTW, I am a licensed plumber, whatever that is worth.
Edit:
To your question about back flushing.........not sure why this really matters as to flushing it, you have to disconnect the hoses anyway, but, some units have a check valve which stops back flow, for various reasons. There are too many different units and brands for me to be specific on yours, but generally, you flush them in the direction of flow, for 30 mins to an hour with a solution for tankless heaters, but most use white vinegar and distilled waters or just vinegar.