Well then the problem is not on the logic board; I think that's a good thing. As I see it, the problem must lie in one of these spots:
Connectors & Cable between Logic board and Analog Board
Components on the Analog Board
The speaker
The first thing I would do is disconnect and reconnect all 3 connections several times to clean the connections. Then I would take something non-conductive to reach in and press around on each connector while playing sounds and see if you hear anything. If so, there is likely a bad solder joint. If that doesn't work, I'd closely examine the analog board and follow where the speaker goes and test/replace components along the way. I can't imagine this being too incredibly difficult to fix even in a worst-case that I can think of if you're willing to do some soldering.
Connectors & Cable between Logic board and Analog Board
Components on the Analog Board
The speaker
The first thing I would do is disconnect and reconnect all 3 connections several times to clean the connections. Then I would take something non-conductive to reach in and press around on each connector while playing sounds and see if you hear anything. If so, there is likely a bad solder joint. If that doesn't work, I'd closely examine the analog board and follow where the speaker goes and test/replace components along the way. I can't imagine this being too incredibly difficult to fix even in a worst-case that I can think of if you're willing to do some soldering.