Make sure you have the right input. My TV has RF, Composite in front, Composite in back, Component, and HDMI. If you do not tell the tv to use RF (antenna), it will go back to the last input source and never see the signal on a different input.
If you turn the system on when no game is installed, the Vectrex will go into Minestorm on its own, You should just sit there with the ship pointing up until the first mine hits you.
Your machine sounds like it is getting a signal to spin the ship even with the controller unplugged. Unplug your controller and inspect the 9 pins in both controller sockets. You are looking for any pins that are bent and touching or possibly any debris like a small wire or metal shaving that is bridging two pins.
See if this helps, as your next step would be to download the service manual and see if the symptoms are listed in there.