I would suspect that you didn't set your GN's output type to "face-corner", which is what kind of data a vertex color layer is.
Notice the geometry nodes sidebar. I've specifically designated the "Color" output as having an attribute domain of face corner. Blender is showing you "face corner" in the properties/modifier tab not because the output is face-corner, but because the type of data to which you're outputting-- a vertex color layer-- is face-corner. Apparently, there is no implicit conversion, which is a shame.
Note that there's a slightly simpler way to do what you want here, which is to just use an attribute node in your shader:
Here, I'm just passing my vertex group through a pass-through, which I'm using to convert from a vertex group to an arbitrary attribute. I've made up a name for my output attribute and typed it in, then referenced that attribute in an attribute node in my shader nodes. This works even as point data, as shown-- it is point data, as vertex groups are not face corner data.