You need to get a manual and give a good read. Then ask your question over with correct part numbers so that people can give you good advice instead of a best guess.

This a much better description of your issues. Certain coils is this game are at high voltage (50 volts) and need a relay to activate the high voltage. Since none of them are working I would expect that the fuse is burnt. If you look below the flipper power supply there is a single fuse by itself with a purple/yellow wire on it. Remove and test it.

When you replace the fuse with the correct size and type rerun the coil test. Stop at each coil for half a dozen pulses while watching the fuse. Note which one it burns out on. Report back your findings.

That fuse will fit the fuse holder but it's the wrong type of fuse, you need a slow blow fuse. You can still use it for troubleshooting.

Here are some places to get the correct fuse.

https://www.greatplainselectronics.com/proddetail.asp?prod=MDL-2-1%2F2

https://www.pinballlife.com/1-14-slow-blow-fuses.html

https://www.marcospecialties.com/pinball-parts/MDL-2.5

https://www.digikey.com/product-detail/en/eaton-electronics-division/BK-MDL-2-1-2-R/283-2743-ND/954323

So now you have a shorted transistor. Its easy to find if you have a DMM. Remove the 3 connectors that I have circled in the pic below.
Set your DMM on resistance, place the black lead under the ground braid cable stapled to the wood in the backbox.
Use the red lead to test the metal tabs of the driver transistors with the power turned off. There are 22 driver transistors that look like the pic below.
Note the number of the transistor when you get a low ohm reading on your meter. Let me know what you find.

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I may have not been as clear as I should of been, you need to test the pins on the board not the wire connector.

This means that the driver transistors are good but a predriver transistor maybe bad causing the driver transistor to turn on when it shouldn't. So retest this time with the power on.

Looks like your testing is fine. Did you test all 3 connectors? Could the ground braid be bad? Try retesting with the black lead on a board screw head and the red lead same as before.

It may not be a board issue. It might be a pinched wire. Leave the 3 connectors off, replace the blown fuse, turn on the power. Does the fuse still blow?

Try reconnecting one connector at a time to see which one will blow the fuse. This will narrow it down to 8 instead of 22.

Magic!!

Maybe a pinched wire.

Check the adjustment of the switch. Change the diode on the coil, make sure to install it correctly. Silver line toward the purple/yellow wire.

You will need to add a diode on any coil that doesn't have one on this era machine. The electrical spike when the coil discharges can do damage to other components, the diode allows this electrical spike to drain without causing damage.

Now your pic of the divertor coils shows someone has put lubricate on the plungers. All of this crap needs to be cleaned off and the coil sleeves need to be replaced. Once everything is cleaned then you need only one drop of oil on the pivoting pins. Follow this service bulletin.

https://www.ipdb.org/files/804/Williams_1987_F_14_Tomcat_Service_Bulletin_no_number_undated_to_resolve_binding_diverter_gate_assembly.pdf

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Your divertors are already wired to the red/white 25 volt power wire.

Something must be moving around when you put the play field down.