Why can't possible install iTune in macOS Monterey

If you mean "iTunes", it was renamed "Music" two years ago, and the non-music functionality was split out into two separate apps, Podcasts and TV. The old iTunes application won't run on Monterey.

Just what "iCloud" are you trying to install?

What exactly happens when you try to install it?

Just to add, all iDevice backup and syncing is done via Finder.