Post a picture of your board.
Did you replace the IC-sockets?
Did you test without the speech board? Put in a jumper at the top of the board.
https://www.pinwiki.com/wiki/index.php?title=File:WMS_Sys6-7_Sound_W1_Jumper.JPG
Did you read the Pinwiki site?
https://www.pinwiki.com/wiki/index.php?title=Williams_System_3_-_7#Sound_.2F_Speech_Board_Issues

Start pressing down on chips when you know sounds should be coming out. Start with the ROM

It's telling you the computer/audio part of the sound board is working. There is a loss of communication from the MPU board to the sound select circuitry of the sound board.
I don't usually venture into Williams support but get your logic probe out and probe the sound select signals at the MPU board connector 1J8 pins 12, 11, 10, 9, 8 and at the sound board connector 10J3 pins 3, 2, 5, 4, 7 respectively when triggering different game sounds. The sound select signals at connector 10J3 of the sound board go into a 4050 buffer chip at IC5. Using the schematics, check that the buffer chip outputs change state when the inputs do.

The sound board works independend from the power supply.
It just needs 9AC with a central tap.
Here is the schematic for that part.

Sound_Supply (resized).jpg

You can power the sound board with a computer power supply if you need to test it on a work bench.
You connect 12V, ground and -12V insted of the 9VAC.

In this case the capacitor being bad could be a problem.
It is for the -12V power supply and is used in the amplifier.
No -12V, no sound.

Sound_Supply_2 (resized).jpg

You can test the sound triggers by grounding the pins at J3.
You don't have to worry about shorting something, there isn't anyting to short at J3.

By the way, is the connector at J1 still original?
I have a problem with my Blackout sound board.
I need to re-pin the J1 connector.
If wiggeling the J1 connector clears the sound, then you know what to do.