Finally got this issue resolved with Esri assistance!
Turns out the place name points layer in the map service I used for the query task did not have any geometry available to the service. So the query parameter to output geometry did not have anything to work with, hence no graphic points could show on my map. My fix was to just use another map service here that does have place name point geometry available. But, I believe the way to fix the layer from the problem map service would be to go back to the source MXD and make sure the SHAPE field is checked for visible (I had it not visible).