I would have thought the problem to be with the main wire to the motor and not the switch.
The switch isn't capable of handleing any load such as the tilt or starter, only the solonoids or relays to operate them.
Have you done a load test on the battery to be sure it is up to snuff?
It really sounds like a poor connection between the battery and engine, like spare said, it will usually be the ground.
Be sure and check the connections, you can also use the jumper cables (if it will fit) up to the starter solonoid and see if that helps.