Thank you for the kind and very thorough response! 
There’s one difference between your version and mine, which I suspect is the root of the problems I’m having: I need to be able to delete and respawn the widgets I’m creating without touching the rest of the UI, so I’m spawning them inside a specifically-designated canvas, so that I can always “reset” that part of the UI by running the ClearChildren node on that canvas.
As illustration, here is the widget I’m currently using to draw the smiley face, it’s just a text block in the top-left corner of the default canvas:
Next, here is the sub-canvas I want to child each smiley to, in my main UI:
Finally, after deriving Center X/Center Y as discussed, I child the widget to the canvas:
Doing it this way never results in the smiley appearing on screen, and my assumption is that somehow when I try to center the widget, it’s actually moving the entire canvas, and always keeping the text block off-screen… perhaps there’s a different object I should be childing the text widget to, instead of a canvas panel?