Cool, so taking J13 off the driver board you eliminated the pop bumper switch from the list of suspects. That leaves us with J12, the ground path for the special solenoids (including the pop bumpers and probably the slingshots). The fact the coil is activating the moment you turn the game on with J12 plugged in tells us the pop bumper is getting a path to ground the instant the game is turned on. That usually indicates a bad/shorted driver transistor and/or pre-driver transistor on the driver board. You can test them both using the diode test on your meter. Looking at the schematic, the driver transistor is Q4, pre-driver is Q3. The lower left pop bumper is labeled solenoid 18, you can trace it on the schematic from the coil to 2J12 pin 4. There should be a blue wire with a red stripe at that connector--that's the ground wire coming from the pop bumper coil under the playfield. The transistors on the driver board provide the path to ground for the coil, and when they go bad, they will often cause a coil to lock on immediately upon turning on the game.
To summarize: the red wire at the pop bumper coil is the +28vdc supply. The blue/red wire on the other lug of the pop bumper is the ground wire, which runs up to the driver board at 2J12, pin 4. From pin 4, the ground circuit for the pop bumper continues to a TIP120 transistor (Q4).
When the mpu activates that transistor, it completes the path to ground for the coil and the coil fires (pop bumper pops). While I'm not necessarily recommending you do this, here's an easy way to demonstrate what's happening on a simpler level. If you remove the blue/red wire from 2J12 pin 4, the pop bumper should stop locking on. If you then stripped the end of that wire and *very briefly* touch the exposed end of the wire to the ground strap in the backbox, it will activate the coil. The transistor is acting like a switch in a way, when it gets activated it momentarily provides a path to ground. A bad or shorted transistor will often "stay on" causing the coil to lock on the moment you turn the game on. Note the TIP120 is the likely culprit but there are other components in that circuit that can go bad in addition to the TIP120 transistor... the pre-driver transistor, the chips before that, the diode on the coil, etc. Good luck!