Ok then, let's go back a few steps. Rereading the thread, the only times you've mentioned using the meter is for continuity testing (beep/buzz test for fuses and switches). I haven't seen any mention of DC voltage readings, only buzz tests. I would check for DC voltage at the red wire where it's soldered to the flipper button switch. It should have approx 43v DC. Just in case this is your first time testing for DC, here's how: set your multimeter to DC volts. Place the black probe against ground and hold it there (there should be a 1/2" wide metal ground braid stapled along the inside of the cabinet, that would be a fine place to place the black probe on), then touch the red probe to the solder lug on the switch that the red wire is soldered to. You should be getting 43v dc there (if I'm reading the schematic correctly).
Forgive me if you've already done that--i just didn't see it mentioned anywhere in this thread yet.
If you definitely have 43v on that red wire, and the flipper switch is definitely making contact when the flipper button is pushed, then there's a problem with the connection from the orange wire to ground, or the contacts on the flipper button switch are shot.
However, i think chas10e is onto something about the flipper button switch being gapped too wide (the contacts never touch when flipper button is pressed) or it may be a bad switch (dirty/pitted contacts). If you have the black and red probes on each of the lugs (red on one, black on the other) and turn on the continuity/beep test, it should beep whenever you push the flipper button. If it doesn't, the switch is gapped too far apart or is misaligned from the flipper button's actuator, or it's a bad switch, or a broken path to ground from the orange wire of the switch. (Again--granted that there's 43v on the red wire)
Sorry if i sound like a broken record. It's just that the flipper circuit is actually a fairly easy one to diagnose, and it could be something you've (or we've) overlooked. We will figure this out!