On a three way switch of that vintage, you select off, ign, and start. Only the selected wire gets powered, which means in the start position, the ignition wire is off (via the key). The way those boats got around that is that they have the starter solenoid feed the ignition circuit while the starter solenoid is engaged. The picture I have attached is for an EFI GT-40 boat, but yours should be almost identical.
In this wiring diagram pin 6 on the 8 pin connector is the ignition wire from the key, and pin 3 is start. You can see that pin 3 goes through the neutral safety switch to the S terminal on the starter relay. Basically when this is energized, it connects all the other terminals together. So normally the 6 terminal feeds the I terminal of the relay, which in turn powers the ignition, but when cranking, there is no power on pin 6, so the relay shorts the I terminal to the battery terminal to provide ignition.
My guess is either the relay is flaky, or there is a loose wire somewhere in that circuit.