I got what I was after, but I want to make sure I have all of the necessary and none of the unnecessary steps so that people can replicate it in the future.
Per this page: http://memhamwan.org/user-coverage-map-how-is-it-made/
I ran (after setting PATH and GDAL_DATA environmental variables properly):
gdal_translate -of PNG -a_srs EPSG:4326 -a_ullr -88.94745 42.17275 -87.9351 41.41801 "ICS Basic 11.png" "ICS Basic 11-geo.png"
I got the lat\long from the KML itself:
<?xml version="1.0" encoding="UTF-8"?>
<GroundOverlay><name>ICS Basic 11.png</name><color>88ffffff</color><Icon>
<href>ICS Basic 11.png</href>
It gave me a georeferenced PNG slightly smaller than the original from Radio Mobile. I added that to QGIS. I went to the properties for that layer and set the Style to only band 3 and transparency to 255 to property view the file. I then ran a polygonize, appending
-b 3 to the end of the command. It gave me a shapefile that ArcMap still didn't seem to like, so I exported it as a KML.
The KML loaded in ArcMap as a polygon.