Before starting with a prime. Look down carb and operate the throttle and see if gas squirts in the throat. If not, bad accelerator pump or no gas in bowl. What you are doing (priming), the accelerator pump should be doing, if there is gas in the bowl. The bowl can be too low because of problem with: float setting, float valve, fuel pump, dirty fuel filter, vacuum leak, carb leak...since in runs OK after starting, sounds like the rest of the system is keeping up with demand for fuel. I'll bet it's not squirting.
As far as the starter goes: read all the advice on troubleshooting given by everyone so far.
