How do fix my no sound problem on my Sony DAV-S300?

I have a Sony DAV-S300 Home Cinema System. It appears to be in fully working order except there is no sound at all, unless you plug in headphones.

Another thing I have noticed is that it no longer makes the noise that sounds like a relay click when switching on.

I have checked all the speaker connections, MUTING is not on, and the test tone can not be heard. The loss of sound affects all functions (DVD, CD, tuner). It's as if the surround sound amp is not being powered up.

I have opened up the box and checked continuity through any internal fuses - all seem fine. Didn't see anything suspicious in there.

Given the age of this system, it's not economically viable to pay a technician, Sony or otherwise, to fix this, but I would hate for it to go to waste.