Hi bcleader,
Re: "I cannot seem to download music to my new(ish) iPhone 14 Pro for offline listening"
Wondering if you are signed in (to your new(Ish) iPhone with your Apple ID:
How to: Sign in with your Apple ID - Apple Support
Apple Support's article: Add and download music from Apple Music
says:
(NB): "After you add music to your library,
you can download it to your device to listen to offline.
Just follow the steps (in the above article) for your device."
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or
You may need to add the new(ish) iPhone to the list of devices that can make Apple ID purchases:
View and remove your devices that are associated with Apple ID purchases – Apple Support
Eg: says
"More about associated devices
Want to add a new device to the list? Just use that device to make a purchase from the App Store, Apple TV app or another Apple service.
You can have 10 devices (no more than 5 computers) associated with your Apple ID for purchases at one time".
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Re message: "This Device cannot be transferred.
This device must be linked to your Apple ID to play content saved offline. This device is already linked and cannot be transferred for 32 days."
"I have no idea what this means",
Seems iTunes may be expecting you to be transferring info (eg: maybe from an old iPhone back-up to the new iPhone) ... or something similar. I could be wrong.
Apple Support's article:
Use the Finder or iTunes to transfer data from your previous iOS device to your new iPhone, iPad or iPod touch – Apple Support
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More info / support, if needed:
Music - Official Apple Support
All the best :-)