Symptoms of a bad fuel pump would be running out of fuel during acceleration. The carb should still have fuel in it when trying to start even with a weak pump. If you can pump it 30 times and not flood the engine, I would guess the fuel bowl is empty from an internal leak or the accelerator pump is worn. Could also be a combination of the two, pump and carb.
BTW, the fuel filter never will be rid of that air pocket, don't use that as an indication of a problem,
Good luck,
Scott
