I'm a Ca Smog Tech. To me it sounds like your still not sure if its a power issue or you have a component breaking down under stress. Because you stated you had replaced a ignition part and it runs better and doesnt shut down as much I would suspect that maybe you have a coil or a relay issue. A relay can get hot and loose contact. A coil can miss intermittantly or just shut down. A bad ground could cause this. Or lack of tuneups could of stressed it to the point that its shuts down. This applys to an ignition module also. Now if your sure its a fuel issue maybe when it wont start spay some brake cleaner in the throttle to see if it tries to start or even starts on brake cleaner. If it shuts down when the cleaner is removed you now knw the problem. If it doesnt start then you know its a spark issue. Not knowing your vechicle its possible your computer ECM PCM has quads for the fuel injectors. These can go out also. Anyways just giving you directions to go.
Good Luck