Yes, you have a fuel pump problem. The one-way check valves in the pump are letting the gas leak down over a few days of sitting and the pump is unable to refill the float bowl after the gas evaporates/leaks out of the float chamber. Once the engine runs for a few seconds the pump finally starts working, and will do so until it goes dry again from sitting for a few days.

Do a volume/pressure/vacuum test on the pump. I guarantee it will fail.
Jerry


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