Let me sum up what we have so far, and tell me if I'm wrong with anything:
1) Hitting the target causes the machine to detect the tilt is closed
2) If you hook up a different switch to the relevant pins, it works as intended
3) It only does this when the target switch, 15, is closed, but not any other time.

I'll have to test tomorrow, but I thought the bally switch mode on the early mpus only shows the lowest number currently activated. So if, say, row three is stuck on, it will only display the number for where row three meets column 1, and not any others. This makes me think that the whole row for 15 (of which the lowest is 07) is being activated at once. Which would be I6 in this case. And if it only does it when the tilt is connected, then you've got a short on the row wire going to the cabinet, right? Is the late hour making me stupid?
I'm curious, when it does this, what happens if you hit the right outlane switch? Does it say 07 then, or 31?

P.S. Also, why do you say the solder is bad on those two spots? I see burning around the pad, from someone using a desoldering iron inexpertly, but the solder itself looks ok....