The ECM is programmed to run the pump at key on for 2 or 3 seconds to "pre-charge" the fuel rail. But it will then take away the pump ground UNTIL it detects an rpm signal. It certainly seems to me that the ECM may not be getting the rpm signal.
The problem is, I don't have SI for this outboard so I don't know the strategy for firing ignition and or running the pump.
There are 2 crankshaft position sensors...items 5 in the link below....and you might check the wiring going to those.
Although, it might take some digging to access the plug connectors and, since you say that it seems to be firing the plugs, I would think that is all good. But it wouldn't hurt to have a look anyway.
The only "cam position sensor" I find in the parts book is labeled "Pulser coil" item 10 in the link below.
I think that you maybe can get to it's plug connector without too much hassle but I don't know for sure. The wiring diagram I have shows it's two wires, one blue w/ yellow stripe and one black.
Do you have a tachometer? And, if so, is it showing rpm when you say the engine tries to run?
About all I can say is that you should verify there's constant 12 volts power being supplied to the pump via the blue w/red (Bu/rd) stripe wire.
You could then try providing ground to the pump's
black w/white stripe (Bl/w) wire while cranking and see if it will provide fuel to run the engine.
But, even if that works, all it does is prove that the ECM isn't providing ground for the Bl/w wire and more testing needs to be done. And that is going to require reading up on the theory of operation in the service manual.
I say that because, in many fuel injection systems, there is also a "clear flood" mode. The operator gives the engine full throttle while cranking to have the ECM shut the fuel pump off to clear a flooded engine.
So, even a faulty TPS could cause the pump not to run also so understanding how the system is designed is important if troubleshooting gets beyond looking for loose or damaged wiring.
Good luck with it and I hope you keep us updated.