Hi there,Have you tested the main transistor and the pre drive transistor(Smaller one) that run that slingshot coil?Did you check the fuse,Pull it out and check with meter.Are there any diodes on the slingshot coil?if so those will also need to be checked.
1.The main transistor
2.The pre drive transistor
3.The coil and the coil diodes(If there are diodes on that coil)The coil very seldom ever goes bad unless it is fried.
All these 3 would need to be tested/checked with a multimeter.Replacing all 3 would be best so you won't be doing the same work over and over again,it is possible you may have shorted that transistor again if diodes,pre driver were shorted/faulty

The slingshot still not firing?

This could very well be a switch issue,The switch blade might need to be adjusted to make contact,Also make sure the plastic piece on the slingshot is not screwed down to tightly.This will not allow the switch to make contact and the rubber ring hardens around the slingshot

Are you grounding the Tip122 collector or the pre-drive transistor collector or ? If it works grounding the pre-drive collector, then I would go back to the logic signals. Have you checked the logic levels with a scope or logic probe starting with the PIA while in Diagnostics->Coil Test? Is the PIA socketed?

I know you've run this down already, but just a quick double-check: By my reading of the schematics, the PIA output driving the left sling is 11D pin 39 going to the 7407 at 11D pin 5 with output on pin 6 then through the resistor array going to the 7402 at 12A input pin 2 output pin 1 then to Q4 then to Q10.

I think it would be uncommon for it to go bad, but one thing you didn't mention is checking continuity across the 4700 ohm resistor in the resistor array.