I was going to come here and post the RetroSix CleanAmp as well. Strange that HHL charges more for it... Anyway, forget about the original audio board from Sega. If you can revive it and it's fine, great, maybe it's good enough. There's basically no reason not to put in the CleanAmp. I'm sure you've gathered the audio fidelity is cleaner and the max volume is about triple. Totally compatible with the original GG speaker.

Your idea to move a good Sega audio board to the project GG, then putting the CA into your own GG is a good solution.

Trying to fix the old one at this point has only one point of merit: to practice and learn diagnosis and repair of that board specifically, should you be interested in it. You mentioned corrosion, so you can do some continuity checks to see if some traces broke. I would also check the inductor/amp to see if that also has a continuous path. If you do get audio but there's something wrong like static, scratchiness etc. clean up the volume wheel since those potentiometers tend to flake. Fortunately there are aftermarket replacements for them if necessary.