You said that when you move the rear point (second one away from the viewer) back away from the first point (closest to the viewer) by hand, the clock runs, and as it is running, the moveable point eventually touches the "stationary point" (or close to it) . That is correct.
I don't know why they don't touch when the spring is wound down. If the "stationary" arm is moveable, move it until the contacts touch. My points were set up that when the spring was just about "wound down" , the contacts touched and the moveable point and arm drove backwards away from the stationary point. In 3 to 4 minutes, the spring was wound down enough for the contacts to touch again and the cycle repeated. This sounds like what is happening when you manually push the second contact and arm backwards.
If you are not getting movement with the contacts touching and power applied, clean the points with a new dollar bill or a slightly rough surface. The black pitting in the pictures can eventually insulate the points enough that they don't pass electricity. Don't use a file to clean the points, it will damage them. Also, in the picture, the points don't look in alignment (face to face). Try to straighten them if you can. As soon as the contacts touch with power applied, the moveable arm should move back rather quickly. That's the "click" you hear if the clock is working correctly. Also be sure to check that you have a good ground on the clock metal base. One post said to use a 9 volt battery to test with- that's a great idea.
If the clock winding mechanism still doesn't work, replace it. It sounds like you could have a damaged gear which doesn't allow the spring to move all the way through the normal distance, or the electric solenoid that forces the moveable arm back is fried. Rik's Corvette supply has them for about $50.