Alright, so I sort of figured out what the issue was. When using the map from the viewer, the MapTip has to be assigned directly to the Graphic, and not the GraphicLayer. Not sure why this is... when using my own map, I can add the MapTip to the Graphic itself, OR the GraphicLayer, and it will work in both cases. For some reason, when using the viewer's map, adding the MapTip to only the GraphicLayer is not sufficient, and needs to be added to each Graphic individually.
I can work around this limitation (for lack of a better word), although I am still curious as to why this behavior exists. Feel free to add any thoughts on that, as I am very interested to hear them!
Thanks for your time!