Mac not listed as Authorized device in Apple Music

See Authorization vs. association - Apple Community for some background to this. Enabling automatic downloads, downloading past purchases, or enabling Sync Library as part of an Apple Music or iTunes Match subscription should cause your computer to be associated with your Apple Account (formerly Apple ID). It is this association that shows in a list of devices, authorizations are not listed anywhere.

Do you have a subscription to Apple Music? Have you used this laptop with a different Apple Account which might prevent it being associated with the current one? Can you download something from your purchase history? You can try holding down option+cmd as you launch Music, which opens the app in its safe mode, and may perform some housekeeping that could feasibly help. More likely there is some preference file that needs to be removed so it can be reset, but offhand I don't know the path for that.

tt2

You cannot AirPlay from a Mac to an iPhone. It isn't supposed to be an AirPlay target.

A workaround for the automatic download issue would be to make a smart playlist with the rule Location is not on this computer. You could then use Cmd+A to select all items in the list and right-click > Download to download them.

tt2