Single ball williams machines of that era don't check the status of the outhole switch when starting a game. It assumes the ball is there and launches it without taking into account its status. Nor does the switch in any way connect directly to the solenoid. That's completely under the cpu and software control.... software senses switch close and reacts to it, by counting down the bonus, advancing to the next player/ball/game over. Next player/ball causes the solenoid to activate under software control. No switch sense=ball never ends (like you're seeing)
Sounds like you have a dead switch. Time to try running the switch test! Put all the drops up and the ball into the drain see if it shows up. If it doesn't, press on it manually or use something to hold the blades closed. Still not showing up, look for broken wires. Outhole switch is likely the very last switch on the chain.... tug gently on the wires where they are soldered to the blades.
Go read pinwiki in the system 4-7 section, switch matrix is covered really well there, including how to test.
http://www.pinwiki.com/wiki/index.php?title=Williams_System_3_-_7#Switch_Problems