Heater smells like burnt hair when it runs-air purifier?

I am going insane trying to remove the burnt hair smell from my air quality. I am in a rental house and since day one it has smelled like burnt hair when the heater is on. For AC, the house has no smell. Once I change the thermostat over to heating, the hair smell comes back. If the heater turns off and then back on, back comes the stinky smell. I have done everything I can think of to remove the issue. I replaced the air filter a hundred times trying to improve the smell. I vacuumed the inside of the HVAC system. I even stuck a scrub brush in the ductwork to do my own ductwork cleaning. I am now thinking I need an air purifier or to get a real HVAC contractor to do the job. The hair and dirt must be deep in the HVAC or the ductwork. A real HVAC technician with good tools could handle the job. What are the chances that my landlord will spring for something like this? I am not thinking likely since my HVAC does work well, it is just smelly. That is why I might go the air purification route instead. An air purifier is not all that expensive. They are made for removing odors from the house. But, odors like food, garbage and smoke are fixed by air cleaners. Can an air purifier fix a burnt hair smell that the HVAC creates? I really hope so since I am desperate. Candles, incense and cleaning the house doesn’t do anything.

gas heater