Does my furnace's air filter do anything at all?
the air filter is designed to clean the air flow prior to passing through the fan or the heat exchanger.  its a common misconception that the air filter cleans the air for us humans.  its there to keep the air clean for the machinery.  its a secondary benefit that the air is cleaner for us.  if you get a buildup of dust inside the fan, you get overheating of the motor, inefficient air flow across the vanes, even fires from localized buildups.  if you get dust buildup in the heat exhanger, it can massively shorten the life of the furnace (particularly in electric furnaces) or cause an outright fire.

if you cant get a cartridge filter to fit your furnace, get an hvac company to come in and install one between the cold air plenum and the furnace intake.  then you can keep the furnace cleaner longer, which will mean a longer life for your furnace.