I was able to get it to work!!! First, I created the ISS recording file just as I mentioned above, using the command:
\\SharePath\Setup.EXE /x /r /f1"C:\Path\Uninstall.iss"
Then I used the command:
\\SharePath\Setup.EXE /s /f1"C:\Path\Uninstall.iss" /f2"C:\Path\Uninstall.Log"
And it uninstalled the software silently just as it is supposed to. Evidently the /f2 parameter is needed to complete the operation successfully. I am very happy I got this to work! I hope it will be useful for other Administrators in similar circumstances!