I figured it out. When you run the ArcGIS authorization utility, it shows you the Authorization Number (ECPXXXXXXXXX) number that it reads from the provisioning file.
The problem was that, by default, the utility was reading the wrong Authorization Number from the provisioning file. It was reading the Image Server Authorization Number to authorize Server instead of the Server Standard Authorization Number. Once my GIS POC sent me the screenshot showing the two different numbers, I realized what was happening and changed the ECP number to match Server instead of Image Server.