Seems like a lean mixture problem, since it works when you prime it continuously. That means either poor fuel flow from the carburetor (as Joe said) or air leakage into the induction system (as Bruce said). Seems unlikely it is a blocked fuel tank outlet: if it was, the primer bulb wouldn't help. It can't be a needle-and-seat problem if a carburetor swap didn't help. (You did ensure that the swap carburetor worked properly on another mower didn't you?)
I think you need to go back to looking for induction leaks - if not into the intake tube, then directly into the crankcase. There are too many possibilities to go into them here. You said you had another mower with identical symtoms: have you made some assembly error on both of them?
A brute force diagnostic technique I've used successfully when desperate is a massive substitution process: swap major components until you find which one the problem stays with. For example, you could swap the whole cowl/fuel tank/fuel hose/fuel tap system. Then you could swap the carburetor and spout as a complete assembly. Then you could swap the base engine as a complete assembly. It has to be one of those three systems. When you know which one, work on that system only.
