That rod may be connected to a hollow tube. That tube is moved by the lever and rod. While sometimes they are a pain to pull out, it should be able to be pulled out after you take off the overflow cover/lever assembly. The rod is threaded with a locknut to allow you to adjust the rest height of the tube so it blocks the actual drain. Sounds like yours, with the accumulated hair/gunk/etc. is not being pulled up far enough to clear the drain line.