I've had the same problem every time I set up a new device (iPhone, Mac, iPad) or I reset the iCloud account on a device. It takes forever for the Notes to sync to the new device while the syncing symbol keeps endlessly rotating.
The solution I've found is the following: the Notes app (both on macOS or iOS/iPadOS) needs to be active in the foreground till the whole sync process is finished this includes NOT locking the device or letting it display go automatically to sleep, i.e.:
-> prevent auto lock-screen in the settings
-> open the app
-> keep the app open
-> NEVER switch to another app or go to home screen on iOS/iPadOS or switch focus to another open app on macOS!
-> wait for the whole sync to finish (which might take a while, ~5-10Minutes in my case)
-> after the sync is finished you could reactivate the auto lock.
Let me know if this helps.
PS:
- I think the reason is that Apple sends the whole history of changes made to the notes one small changeset at a time and applies them one after another with some some idiotic implementation somewhere, and whenever the device is locked or the Notes app is not in the foreground the whole process seems to stop or potentially start all over again (the same happens, if you get frustrated and reset your iCloud). This might not be a problematic implementation if one has only one note for grocery shopping, but if you have many big notes that you've used for years, this will take forever. Great work Apple!!