I think the reason this is a mystery to you is you've not really grasped what Blend Modes are in general, and Overlay in particular.
Anything in Overlay mode is invisible against black (or white). Change your black layer in the PSD to any color or a dark grey and you'll see the white outline. Overlay doesn't affect pure black, nor pure white. You can demonstrate this by painting with a black brush in Overlay mode on a white layer, or a white brush in Overlay mode on a black layer: nothing happens. That's just the way the math of Overlay works.
The problem, and the thing that's confusing you, is that your Drop Shadow only goes into Overlay mode when you turn on the layer below. When that layer is off, it's not in Overlay mode, it's in Normal mode.
Blend modes are blend modes. A "blend," by definition, requires at least two ingredients. In Photoshop, a layer or an effect blends with something below it. There must be at least one non-empty layer to blend with, otherwise the layer or effect displays in Normal mode. That's why you can't assign a blend mode to a background layer: with nothing below it, it can't ever blend.
Set any Photoshop layer to Overlay (or any blend mode). If you have no layers or only empty layers below it, you will not see any change; the layer has nothing to blend with. Add a layer below and fill with a color. Now you will see a change, because Photoshop now has a second layer to use in the blend calculation.
Your PSD clock has a white drop shadow. You should now have realized that it is a white drop shadow in Normal mode, regardless of what blend mode you assign to it, until you turn on the non-transparent layer below it. Because you're using a black layer, the Overlay mode white simply disappears in the blend.
PNG is a web format. It contains transparency, but browsers, HTML and CSS don't understand blend modes so there is no reason for a png to contain that information. When Photoshop exports your clock-with-transparent-background to png, it exports the white "shadow" in Normal mode because it is in Normal mode, and because any blend mode information is meaningless to png.