Q: how do i get the music into iTunes? after backing up my iPhone on a new hard drive, how do i get the music into iTunes?

Your i-device was not designed for unique storage of your media. It is not a backup device and media transfer was planned with you maintaining a master copy of your media on a computer which is itself independently backed up against loss.  To use a device with a different setup you transfer the old library from a computer or a backup directly to the new setup, not the device to the library. Media syncing is one way, computer to device, updating the device content to the content on the computer, not updating or restoring content on a computer. The exception is iTunes Store purchases which can be transferred to a computer.

Redownload or transfer your iTunes Store purchases from an iPhone, iPad or iPod to a computer - https://support.apple.com/en-us/HT201267 - "This feature works only for content bought from the iTunes Store." As of iOS9 it no longer does apps which now must be re-downloaded directly from the Store.

For transferring other items from an i-device to a computer you will have to use third party commercial software.  See this document by turingtest2: Recovering your iTunes library from your iPod or iOS device - https://discussions.apple.com/docs/DOC-3991  Even this method may not fully recover what you had in the library originally. For example if in order to save space when syncing you had converted music files to a lower bitrate, or photos to a lower resolution, it is those lower quality files you will recover.

If you subscribe to Apple Music, tracks which are not part of content you have purchased or uploaded cannot be transferred and have to be downloaded directly from iCloud.

Yes, discussed in my third paragraph above.  I don't know how up to date that list is and you will probably just have to judge for yourself and see if there are others that work better for you.