I'd start by checking voltages at the power supply board. Without good voltages, you definitely won't get the game to boot.

Was the game working previously, or did you acquire it in this condition?

I would reset the boards and connectors and see if anything changes. Report back.

I recognize this Gorgar.

What was the last thing that happened before it went dark?

Pinsiders, machine history is that it was bought at a yard sale non working. Boards were rough, all boards were fixed by one of the pinside board pros. Fixed boards got the game running. Ball feed, slingshots, bumpers, target bank resets were still out. Replacing F4 (2.25A sloblo) got the rest running.

We rebuilt left flipper, which was binding.

F4 blew out again the next day, and now it seems to have gone dark.

The coil that moves the sling does not have a light built in. If the bulb under the plastic has burnt out, replace it. If your actual coil was on fire it would be bright for a short time, then slowly get brighter as your entire machine burned to the ground, then the whole machine would be blackish.

Seriously though, what went black? Lights under the plastic piece covering the sling are part of the G.I. general illumination and if it's not just a burnt out bulb there will be a whole string of them out and a fuse specific to the G.I. circuitry.

Have you done what frunch suggested a couple days ago? It looks like your MPU isn't booting. Pinwiki has a ton of information: http://www.pinwiki.com/wiki/index.php?title=Williams_System_3_-_7#Quick_Start:_Repairing_a_Williams_System_3-7_MPU
It would be worth asking the person who did your board work what upgrades and other work they have already done so you don't waste time duplicating stuff they already covered.