There's somebody else on Pinside with a Genie with a similar problem. On the 'cardboard' thing, the guy said it was missing from one coil and that was allowing the lugs to short to the coil mount. Normally there is plastic around the lugs there to prevent shorts but maybe his had been broken off or something and someone stuck cardboard between the coil base plastic and the mount as an insulator.

Only place I might put a tiny smidge of lube is where the cams activate the switch arms since it's metal to metal but doubt it would have any effect on your problem one way or the other. Strange the way it works 100% with any 6 targets, but one more and they barely move. I would think with 7 it would at least raise them pretty good but might not be able to latch a few of them.

You said you tested the coils but are you sure that both solenoids are indeed being powered when it resets, and not just one? Also wondering if each of the two coils is powered by its own driver transistor, perhaps remote ones, and that maybe one is kaput.