Using Solid State Relay board for wired low-voltage switch activates AC outlets
Yes, with the clarification I think that would work.  Stereo closet has the mains power, one end of speaker wires and one end of a Cat5 cable.  Remote location has other end of speaker wires and other end of Cat5.  Low voltage power source (5-12v) applied to Cat5 in stereo closet, switch closed in remote location on Cat5 completes circuit and triggers relay, activating mains power to room amplifier.  Input from audio source amplified by room amplifier and piped to remote location via speaker wires.

I think that you may want to consider a latching relay.  A regular relay makes a connection when the coil is energized.  You hook up the thing you are powering to the NO or NC contacts and when the coil gets power the thing you are powering is switched on or off as long as the coil is energized.  A latching relay would allow you to leave the room amps on with the relay in a resting unenergized state.  A regular relay is analogous to an invisible somebody standing there smashing down a momentary button for as long as you want the amp on, while a latching relay would be more the passive switch I think you have in mind.

Finding a suitable predone 8 channel, zero crossing, latching relay module could be a tall order.  You may need extreme Google-fu or Digikey-fu.  Or 8 1 channel models. 

HTH.