You are close.
If the motor has voltage to the hot side. And a clean ground on the ground side it will turn, if the motor is good.

Make sure you just don’t have voltage but a clean ground,

Also, the relay board can cause problems. Make sure this fires, and gives voltage across the relay. I have seen where these don’t fire, or fire but have bad contacts that don’t let voltage flow.

Do the motor test, just with jumpers. If you are not comfortable doing this get someone to help. If you do wrong your problems will get larger

Hello,
Having the same issue. Island will work at times after the machine is off sometime. Was working fine, but just quit. Did some investigating and watched a video on YouTube made by TNT Amusements on a refurbished machine. They stated on the video that they had got one of the early Gilligan machines (back when they were new) and found out from the factory the green underside of the rotating island causes a reflection and gave issues for the opto to read correctly. They recommended to paint the underside black. I am in the process today 5/27/2020 doing that to mine. I am doing it in a flat black so that there is no reflections. I am hopping this will fix my issue of the island not working. Might want to try this to yours too. Will report back and let you know.......Mike

It should fix the issue! Todd

Well fixed my issue and here is what I did. Painting the underside did not fix the problem. The island worked and then didn't. Re-seated all connectors and ribbon cables associated with the island. This did make the island work, but not for long. Re-seated the connectors on the relay board and it worked again, but then didn't. Removed the relay board and found that it had many cold solder joints. Re-flowed all points and this repaired my issue as the island now works flawlessly. Hope this helps you with your issue on your island.....good luck....Mike