If you stick a voltmeter on the leads of the coil to the ball eject, do you ever see a spike of voltage when it should be firing?

Use jumper wires with alligator clips to ensure a good connection. Clip them to the coil leads and then to the test leads off your meter. Set it top the V-(the V on the left) on the 200 setting. That's DC voltage in the range you are expecting. Fire up the game and watch for a spike in that voltage. If it stays zero and doesn't change, it isn't the coil. Another test would be to pull the leads off the coil and plug them into another coil, like one of the slingshot coils. If it fires, I'd look at replacing the coil.
However, something did occur to me. Does this machine have a trough switch that would prevent the eject from firing if a ball isn't present? If so, that switch could be defective. I would test that by connecting the switch leads using a diode and seeing if it changes the behavior.

It's probably a 1N4004 or 1N4007. Either should work.

Who is it that is refusing to sell a coil? Just go on Marco or pinball life and order one. Seems odd. Anyway, if it were me, I would want to know if there was electricity getting to the coil in the first place. So, I would attach the connectors to another coil, like a pop bumper, a knocker or something, and see if that activates(pro-tip: keep a spare knocker around for this purpose. You don't have to be looking at it to know when it activates). If it doesn't, the problem is upstream from the coil. Look at the wiring and the controller. I don't know how Brunswick controllers look, but I imagine that it's a simple transistor circuit.

If it does, well, it's got to be the coil or the diode(assuming that the physical mechanism isn't bound up).

Well, I can't find schematics on the actual controller. The red wire to the coil should be 28v. Find a ground point somewhere and use your multimeter on the setting I mentioned above and see if there is a constant 28v. If that doesn't read 28v, then power is not getting to the coil. Trace the red write back and see where there is a break.

If there is 28v, set the meter to test continuity(bottom right yellow 200) and put one lead on the other post(the black wire) and pin 15 on connector CR-1. If it doesn't drop to below 1, then there is a break between those points. If it does, then it means that the problem is on the board. Without a schematic, I can only tell you to start at pin 15 on the connector edge and follow it until it hits a MOSFET or other transistor or relay.
Can you post clear pictures of the front and back of that board?

So, this got me thinking. If the problem is on the board and isn't a simple component replacement, you will need to bypass it. A simple, if inelegant, solution would be to run a manual switch to trigger the ball eject.
Otherwise, this is a simple enough game that you could convert to another controller, like an arduino with some MOSFET boards, or the OPP boards.

Yeah, but that ends at the board edge. Does anyone have the schematic for the board?

To bypass should be easy, if it comes to that. Run the black wire to a momentary switch and from there to a ground. Another possibility(more ambitious) is to add another switch with a microcontroller to see it and pop the solenoid.

Well, it looks like pin 15 leads to that big power transistor, 2n6576
I would replace that if you don't find a problem with the wiring.

Yeah, it should be fine. That's a manufacturer's code.

Also, yes, replace the diode as well. Buy a few extra, as they are used a lot.

Yeah, check voltages first. If you happened to nail a chip with some static electricity, it could have died. But I would recheck all of your connections. You might have something unplugged or backwards.