A very common mistake coming from people wiring up their own motors is to hook up the purple/white wire in the harness to key on hot power. Should be the purple wire on the "I" terminal. If this has been done then the primer solenoid assembly is toast and fuel will free flow through the assembly. With the engine off and the key on you should be able to hear crisp audible clicks with you pushing in the key switch repeatedly. If you do not hear these clicks then the primer has failed, ground wire is open, or it has been wired incorrectly. Quick checks to make is to take a pair of needle nose pliers and gently pull one of the small diameter hoses off the primer while the engine is running or you are pumping the bulb. If fuel is coming out then take a test light and see if the purple/white on the primer is hot while the key is on and the key switch is not depressed. If so, go to my first sentence and fix the problem. Second way to check is to pinch off the larger line to the primer while running and see if the engine picks up. The primer does not stay on until a certain temp, the Purple/white should only be hot when the key is on and the key switch is depressed.
Third..........the red manual valve should be facing back along the body of the primer, not sticking above it.
Below is how your key switch should be wired. There is no need to add a push button switch!
IDENTIFIER FUNCTION WIRE COLOR CODE CONNECTS TO
B BATTERY Red with Violet stripe Battery positive via fuse
A or I ACCESSORY or IGNITION Violet Accessory loads; engine ignition
S START Yellow with Red stripe Engine starter solenoid coil
C CHOKE Violet with White stripe Engine primer solenoid coil
M (raised) MAGNETO Black with Yellow stripe Engine ignition kill circuit
M MAGNETO Black Engine chassis or battery negative