Bad 40-pin connector between the CPU and driver. All sorts of open/cold solder joints at the .156" connectors. A great place to start is this thread:

https://pinside.com/pinball/forum/topic/vids-guide-to-bulletproofing-williams-system-6

The MPU sits in a u-shaped channel behind the driver board. You will have to at least remove the screws holding both boards, then you could separate the MPU and remove it.

The issue sounds to me like one of the chips for the switch matrix lines on the upper right corner of the driver board is bad.

I bet it's not "seeing" all 3 balls. Go to switch test and check/clean all the trough switches.

Edit: didn't see the door switch not working part.

The switches in the upper trough often need adjusting. If there are balls there at the start of a new game that could be culprit.

If the game is booting into audits, you have a bad battery or bad RAM on the CPU board.

Reflow the solder on the diagnostic switch connector on the CPU board or replace the header pins entirely. If that's been done, replace the IDC connector with a new crimp on style connector.

It's to the left of the 2 PIA chips positioned on the right side of the cpu board. I screen capped an image of the cpu board and drew an arrow pointing to the 5101 RAM chip. You may want to replace it with an NVRAM chip so you can eliminate batteries entirely.

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Did you carefully inspect the battery socket to make sure the contacts are not corroded? With the game off, measure the voltage between pin 8 and 22 on the RAM chip to make sure you have 4.5 to 5V

These chips are VERY static sensitive. If not careful handling them they can easily be damaged.

You might have a bad socket there. Old single wipe sockets don't last forever.

I have a Black Knight with the exact same problem.
Not sure if my fix will work for you but see if it does.
I was working on my Black Knight and had a ball in the shooter channel so I took it out and put it in my pocket. Turned on the machine and the symptoms were just like yours.
Realized I had the 3rd ball in my pocket and put it in back in the drain hole. Turned it on and everything started working. I had no idea the game does a check for 3 balls when it turns on. There might be 2 in the upper deck ball lock and 1 at the bottom. Whatever the combination there has to be 3 balls in the machine at all times or it won't start a game. It will do everything short of letting you hit the Start button.
I also was given extra balls with my Flash game so I put one in the Black Knight. Turns out it was a smaller size and thr kicker would not stop trying to eject the ball into the shooter channel. I had to turn the game off and physically remove the ball.
Also don't add more than 1 ball to Flash. It freaks out if there is more than 1 and will keep trying to add a ball to the shooter channel.
Hope that helps!

P.S. - Your backglass looks amazing!

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