The Ball Release is solenoid 1, that's Q14, Q15 and part of IC1. You can try a quick test by clipping a jumper to the tab of the transistor and then touching the other end of the jumper to ground. If the coil fires, it's a bad transistor most likely.

You should also check the type and rating of the F2 fuse. With the coil cooked that badly, I think it should probably have blown. Make sure it wasn't overfused if your other solenoids are still working. If they aren't, the fuse is probably popped.

This is good advice. The jumper test simply tells you that everything else down to the coil is good and the transistor isn't going to ground as it should. This could be because the transistor is bad OR those components that drive it are bad. Since you've replaced the transistors, there is a high likelihood that the portion of the 7408 that drives that transistor is bad. I'd pull it, add a socket and plug in a new 7408. If you have a logic probe, you could probe the associated pins on the IC to see if they are active but replacing is pretty easy and cheap.