Then yo do not have GSX installed, must be in X:\Addon Manager
The default installation for ALL FSDT products ( sceneries for FSX, P3D and MSFS ) and GSX is:
C:\Program Files (x86)\Addon Manager
Of course, if during the first installation the users chose another place, which could be another drive too, that might be anything, like D:\Addon Manager, E:\Addon Manager, but, regardless of the actual location, the FSDT Live Updater executable, which is named "Couatl_Updater.exe", will always be in the root of the Addon Manager folder.
A Desktop icon would be created automatically by any installer but, when uninstalling something, the Desktop shortcut will be removed, but the executable will still be there, so it can be launched manually, it won't make any difference if there was a Desktop shortcut or not, which would be recreated by running any installer.