Using an op amp as a switch for audio (Dual supply +-12V)
Try this one for size: -



I'm not saying it will exactly suit your application of course but it does provide food for thought.

Alternatively you should be able to use an analogue switch at the output. You say: -


  I'd rather not have to bias and normalize the signal in order to use a
  typical analog switch


This doesn't really make sense because you power the A-switch like the op-amp with dual supplies like this one for instance: -



It's a quad package, runs from +/- 15 volt rails and has a control signal that is 3 volt logic compatible. It cost £1.76 at 1500 pieces from Mouser so if you have 4 channels (as implied in comments) then each channel costs £0.44: -


  @Andyaka opa1679, it's a quad channel, there are 4 inputs and I would
  like to be able to turn them all on or off more or less simultaneously


One quarter section can both disconnect the op-amp output and provide another line where you can connect a pull-down resistor on the output.