Welcome to the fun.
I'm no expert but I have just been through this ride, this is just my approach.
It looks like the machine has had issues and somewhere along the line the MPU board has been replaced, from what I can find online this should have a system 6 board but your one seems to have an upgraded system 4 board installed.
See section 3.3.6 here on page 13...
http://www.pinballsupernova.com/Williams Repair Guide/Williams System 3 through 7 .pdf
It is not the end of the world, before all hope is lost we must assume the game once worked with this configuration. Question is do you want to play the needle in the haystack game or get someone to fix it for you ?
If you are willing to go on the pinball fix it adventure and find the needle I would do the following, PLEASE BE CAREFUL and know your limits as the last thing any of us want is a human casualty.
The schematics form here...
http://mirror2.ipdb.org/files/1413/Williams_1979_Lazer_Ball_Full_Manual.pdf
...and Clays guide as mentioned above will be invaluable.
As it sounds like you have some lighting on the playfield it seems power is getting from the wall into the machine ok. The next thing to check is the outputs from the transformer. The transformer should be in the bottom right corner of the head with the boards in, there should also be a ground braid wire that comes up from the lower main cabinet that is terminated somehow in the head box. If the ground is connect proceed to check the voltages on the the pins in the two plugs that come from the transformer and plug into the board above it. Note that these tests will be in AC and that there is usually a high voltage output ranging from 80 to 120vac so again please take caution. I believe a +/- 10% tolerance is allowed so readings may not be exact.
According the the schematic you should have...
White to White wires - 90VAC
Blue to Blue wires - 13.5VAC
Red to red wires - 25.5VAC
Grey to Grey to Grey wires - This is more tricky but across two of them you should get about 9.3VAC, other combinations will give you less.
Yellow to Yellow - 6.3VAC
Next with the power off, plug the transformer plugs back into the power supply board and check the fuses, make sure fast fuses are fast and the slow are slow. also make sure the screws holding the power supply board into the metal brackets are all in place and secure.
Use the continuity test on your multi meter "buzz" to check the power supply board is grounded to the ground braid from the lower cabinet, this is a backup measure. The boards ground trace is where the securing screws go through. It looks from your photos above that you already fixed this braid back to the lower cabinet. It may be worth (with the power off and plug out the wall) buzzing the ground braid repair to the mains ground to make sure its true.
Remove all the perimeter connectors from the power supply board and turn the power on.
Ground your meter to the ground braid from the lower cabinet and test the DC outputs on the power supply board 3J6 connector as per page 8 from the schematics linked above.
Once you have ascertained those are good you can move on to diagnosing the MPU board and see if its playing ball.
Regarding the general state of the boards, its up to you how far you want to dive in but there is an informative thread on pinside that may be of use (
https://pinside.com/pinball/forum/topic/vids-guide-to-bulletproofing-williams-system-6 ).
Happy fixing and be safe.
Guy