You have to use scripts to change anything other than keyframe-able layer and effect parameters. You might be able to fake a "click" effect with an expression by triggering a momentary change in the volume of a constant noise audio layer, and using markers to trigger it.
I think using the start times of other layers is problematic because writing an expression that would check any number of layers would involve some kind of for-loop that could get complicated, and you can't easily pass values or variables among different expressions. The question with expressions in AE is always whether the solution saves you time in the long run over just doing it manually, so it depends on your needs.
The quickest way to do it would probably be to just pre-comp your sound effect and whatever layer it needs to match, so that each time the pre-comp plays, you also get the click.