I don't own an STTNG but whatever shot starts that mode must have a flaky switch, mechanical or opto. It obviously thinks you've hit that shot and so it's starting the mode.

No, nothing to do with the driver board. Everything to do with the switches related to that mode.

That's a weird problem. I don't think there is a particular switch that starts it. It either starts from Choose Mission, or when it's "time" comes. What I mean is, in between balls the next mission will flash until you shoot Start Mission to actually start the mission

I almost wonder if he has a faulty ROM

That seems logical.

Can you rephrase this sentence in a way that is less ambiguous?

I can interpret it as "The ball will be directed to the left cannon when it shouldn't be" (which per the rule sheet, will allow you to shoot it per the normal rules)

Or "The game will behave as if the ball is in the left cannon, when the ball is not"

In the second case, the problem is a switch problem. ST:TNG cannons are notorious for that.

In the first case, the problem is likely with the mechanism used to feed the balls to the left cannon; check the applicable diverter mechanisms.
Depending on how they work (I havn't gotten to see a ST:TNG's internals), this could be anything from a jammed mechanism to a loose wire/bad coil (if the machine needs to fire the coil to direct the ball to not-left-cannon) or blown (stuck on) transistor (if the machine needs to fire the coil to direct ball to left cannon, and the transistor is stuck on, it would leave the coil permenantly energized and hence always on). Basic examination of this assembly should make it clear which of these is the case.

Check the lower diverter (the one that chooses left cannon or lobs it onto the habitrail leading to left inlane) - finally found pics of the underside of the PF (why aren't there any pics on Pinside?)