Wouldn't have mentioned my situation, as I am running a preview version of Win11, but it is notable that this problem isn't just associated with a single Win11 build.
iCUE 5.1.1113
Win 11 Version 22H2
Build 23440.1000
Availability date 2023-04-14
(With the realization that it's obviously possible for more than one person to encounter the same installation issues) I just wanted to point out that I am having the same problem with Windows 11 and the iCue system tray icon.