When you hit the 'test' can you tell that it is cycling sounds? You may not be able to hear the sounds themselves, but should hear pauses in the static etc.

Is the white noise constant or only when a sound plays?

It isn't going to be the sound chips and it sounds like you already ruled out the pots and amplifier.

As the others mentioned, if they haven't already been replaced, do the caps first and see if that fixes it.

I'm not 100% sure if that cap being bad would give you your symptoms, but it is always highly suggested it be replaced. If you ordered one and didn't get one, contact him and get one sent. I've bought a few of these sets from him over the years.

If you turn both pots all the way up, are you hearing any high pitched noise pulsing? (just asking because the one I worked on had this symptom), but that one had multiple things wrong with it. If you looked at that thread, there's a bunch of threads here that go over this board as many people have had issues with it. Yes there's a bunch of 'noise' to filter out, but the eventual fixes are described.

Things you can do for free and not wait for parts:

Make sure all your socketed chips are pressed in completely on both boards. I do not recommend removing them as a first step because the pins on these old things can be brittle (which may be something that needs looked at as well).

Go through and reflow the solder on the header pin connections on the back of the boards as well in case of cracked solder.

Validate the output to the speaker cable is connected and wires are good?

Additionally, if you have a spare IDE cable laying around, swap the cable that connects the boards and see if that fixes it. Without both boards talking to each other you won't get any sound.

Unless you have a logic probe at the least, or really know your way around the schematics and voltages, shotgunning easy stuff first is your best bet.

If you need expected voltages of each chip pin, I can help you there. I graphed out every pin on a working vs. nonworking to track down an issue. Really not enough information for me to say if it is on the digital side or analog side...

I still maintain to change out that last biggest cap though.