How to remove "Start using iCloud" notification badge in System Preferences?

I also ran into this issue. I've signed-in to the App Store but didn't want to enable any other iCloud services.

It turns out you can remove this badge from the command line with

defaults delete com.apple.systempreferences AttentionPrefBundleIDs

That said, you might want to check the value of AttentionPrefBundleIDs before deleting the key. If there's anything other than "com.apple.preferences.AppleIDPrefPane" present, then re-writing the list is probably better ("man defaults" to read how to do this)

$ defaults read com.apple.systempreferences AttentionPrefBundleIDs
{
    "com.apple.preferences.AppleIDPrefPane" = 1;
}

$ defaults delete com.apple.systempreferences AttentionPrefBundleIDs

That's a tough call. I certainly understand and respect your unwillingness to accept Apple's Ts & Cs but I just don't happen to know how to disable the nag without enabling and subsequently disabling iCloud. I also understand you can't even get past that step until accepting Terms you are unwilling to accept.

The following Discussion seems to confirm that suspicion: Catalina: How to disable iCloud? - Apple Community

It seems to me that if you don't want to accept the Terms then you ought to be able to disable the nag without accepting them as a prerequisite. If I were you I'd contact Apple, express your objection to that (apparent) practice, and then request they provide a way for you to turn off the nag. Start with Contact Support and click through the options until you find a way to speak with a human being. You will almost certainly need to speak to someone other than the unlucky soul who happens to answer your initial call, who will only be skilled in reiterating the fundamental steps you already know.

If there's another way around it I don't know what it is. I suspect modifying an existing User Library .plist file may do it. If that's the case an Apple Support representative would know which one and what to do with it. They are plain text files.

I think all you need to do is open System Preferences > iCloud, and respond to whatever it wants. You can subsequently disable it, or sign out of iCloud as you prefer. That should terminate the persistent nag. I don't have a way to verify it though.

Note that you'll have to exit out of System Preferences and re-open before you'll see the change.