Not all messages appear in iMessage after setting up new iPhone and restoring from iCloud backup

There is some confusion between iCloud sync and iCloud backup. I’ll try to clear it up:

iCloud+ has 2 independent functions. iCloud backups, and iCloud sync. It’s confusing because they both have the same name prefix. But they have no connection with each other. Go to Settings/[your name]/iCloud. You will see a bunch of switches (with iOS 16 or later also tap Show All). When you turn on a switch that data type will sync to iCloud. If you do this on multiple devices that share an Apple ID the selected data will sync to all of those devices. 

You can also turn on iCloud Backups. This will back up your phone every night if the phone is plugged in, connected to Wi-Fi and locked, and you can also back up manually.

But note that these are independent functions; anything that you sync by the first method will be excluded from the iCloud backups, and vice versa.

Was the phone connected to power and Wi-Fi? Or did you have sync over cellular enabled? Syncing will pause if the phone is in low power mode, the battery is low, and a few other situations.

Then it should have synced. I can’t guess as to why it didn’t.