Re: primer solenoid problem?
Your motor will be very difficult to start if the primer solenoid does not work. Do you hear the primer click when you push in the key? If so, try this test. Remove one of the primer hoses that leads to one of the carburetors. Pick the easiest hose to re-install. Pump up the bulb and press in the key. Fuel should shoot out the hose. If it does not and your primer is clicking, you likely have an obstruction either in the primer or in the hoses leading too or from the primer. It is pretty easy to disassemble the primer and see what is wrong. Just remove the four screws.
The schrader valve does not really do anything for your priming action. It is used to squirt fogging oil or engine tuner into the engine instead of squirting it into the carburetor barrel.