I looked up this model engine - it says this: "Prime 'N Pull Easy Starting System ™ pumps the right amount of gas to carburetor for easy one-pull starts". Prime 'n Pull.
I have an old Weed Eater lawn mower with a 4.5hp Techumseh engine. It has a primer bulb - is that what "Prime 'N Pull" is referring too? Mine was very difficult to start - either... I finally tracked it down to a small hole in the primer bulb. This effectively negated the choke. Trying to start a cold engine with no choke. I replaced the primer bulb and Voila.
While was trying to find the issue, I could get the mower to start with about 20 primer pushes. And i could keep it running by holding in the primer bulb (allowing the choke to function). Good luck man.