I have the tool you are refering to. I never once had to use it, I always raise the rt side of the car and bleed as stated above and never had a problem. Not just my car but the hundreds I have worked on. I bought the tool for Jeep 4.7 and 3.7 motors, those go airbound everytime you drain and refill.
However I am at my wit's end so I will give it a try. Mike, do you think I should completely drain the system? Which cap opening are you attaching to?