I've searched superuser and Google, but it seems there is no way to show/hide a specific layer in Photoshop using Actions.
Is there a way to use Actions to show/hide one specific layer in a Photoshop document?
No there is no such way. What you could use is called "Layer Comps". There you can save layer position, opacity, show/hide state etc, as a way to show off different presentations of the same content. You might be able to use that for your purposes.