Found! (Thank you Frame Debugger !)
It turns out that Unity renders the Depth Texture in a separate pass before all geometry is rendered. It is hard to find how to include objects in this pass, and how to specify how they are drawn.
The documentation is kind of vague, but it tells you that the "RenderType" is important without specifying what it should be in order for objects to be included in this pass. I found out that a SubShader will render if it specifies
"LightMode" = "ShadowCaster"
So, my shader needed a separate SubShader that also performs an AlphaTest with this LightMode
set. This in addition to the 'regular' SubShader which will draw the sprites; which might not even need an AlphaTest since the zBuffer is not used for the DepthTexture on my setup. The documentation suggests that some hardware/OS/rendering setups will use the zBuffer but it's not entirely clear.