Before you change any more parts, run it off a portable tank. Hook a 6 gallon portable straight to the engine, bypassing the boat's line, primer bulb, separator and antisiphon valve. If it stays running, the problem is on the boat side, and the tank vent is the first suspect. A partially blocked vent lets it start and idle fine, then pulls a vacuum in the tank a few minutes in and starves it. That fits what happened out there... runs, bogs, quits, gets towed. Next time it dies, crack the fuel cap loose. If you hear a hiss, that's your vent.
If it dies on the portable too, then it's engine side, and you need to say which engine this is. In 2004 you could still get the carbed 2 stroke as well as the EFI 4 stroke, and they fail differently. On the EFI, the VST has its own fine screen and racerone is right about pulling it apart. While you're in there, take the idle air control valve off and look for carbon. A stuck IAC will start on the primer fuel and then stall the second it tries to come back to idle.
One more cheap check: after it dies, feel the primer bulb. Soft and collapsed means you're pulling air from somewhere upstream.