As a physician, I strive to understand the underlying mechanism of any physical concern or complaint and to solve problems. In that context, until recently I was frustrated by the apparently simple problem of cleaning our stainless steel coffee carafes of the oily residue that adversely affects coffee flavor. Stainless steel hardly resists staining from coffee. Normal soap and water did not work well, brushes were hard to manipulate through the neck of the carafe, and it did not make sense to have to use an abrasive to remove the oils, which might well damage the pot.
Science Provides the Answer
It turns out that the correct solvent for coffee oil is the solution resulting from dishwasher detergent powder, which contains sodium carbonate. Really simple. No abrasives needed. Any powdered detergent, such as Cascade or Finish, will work.
Put a small amount in the container for the coffee grounds (1-2 tbsp.), run the coffee cycle for a full pot, and you are likely done. If the stainless pot is truly soiled, you may need a second cycle with detergent. Then rinse pot carefully or run a coffee cycle with nothing in it. Enjoy a sparkling stainless pot and fresh coffee taste.