If the choke is automatic and it is closed when the engine is cold and not running, it is most likely OK. If the engine runs properly once it starts, there is probably no problem with the basic carburetor or the fuel system. It sounds as if your problem is with the priming system. Since they say prime it 20 times, they seem to be expecting that the priming system itself will lose prime, and you are supposed to pump air until it finally manages to achieve a liquid seal in the pump, after which it will work properly until it dries out. In other words the priming system is probably crummy, and the manufacturer knows it. This is one of those situations where you are likely to have to either fit new parts to the priming system after each week of actual use, or change the design of the priming system so it works better than the original one. Neither of those options seems all that convenient - I hope the generator was very cheap.
It is difficult to help you with a redesign of the system without seeing the machine. Difficulty priming the primer is likely to be due to one of two things: fuel level well below the primer, so it is difficult to prime the pump; or poor quality priming pump, so it is not good at pumping air. One thing you can try immediately: make sure the fuel tank is completely full whenever you try to cold-start the machine. This reduces the distance the fuel level is below the primer.
