Is there a way adapt the headphone output audio and use it as a switch?
So that a peak audio signal appears at the jack it will trigger a DC circuit.
Is there a simplified approach, without the use of computer chips, etc?
You can use a rectifier and a filter capacitor to detect peak audio output voltage e.g. by a Schmitt-trigger circuit. The output from the Schmitt-trigger can be used for various purposes.
Why does it have to be an audio output? If it's a PC you're talking about, you can use one pin of the parallel port (if there is one) or use a simple USB to parallel converter to achieve the same.