Transferring music from old iPhone to new iPhone without having to re-tag everything manually

So, I listen to classical music, and I don't use streaming apps such as Apple Music or Spotify, I use these old-fashioned things known as files that are ripped from original CDs or transcoded from digital downloads, stored on my computer, and then transferred to my devices for use.

iTunes (and its successor applications I think), ever since Works and Movements were introduced, plays back classical pieces more or less correctly: all of the individual movements making up a work are played in the correct order, regardless of whether or not shuffle is activated. The Music app on an iPhone does not. It ignores Works and Movements (and Groupings, etc), except for displaying them correctly with the work as a heading and the movements as subheadings.

On my old iPhone I used an app called CS Music which, although it also ignores Works and Movements, does recognise Groupings. This makes it possible to play back classical music correctly, with or without shuffle, through re-tagging everything (i.e. transferring the "Work" tag into the "Grouping" tag). However, since I already use the "Grouping" tag on iTunes itself to list the record label, I had to re-tag all of the files once they were already on my iPhone. This means that the files on my iPhone are the only ones that can play back in the correct order on an iPhone.

I recently bought a new iPhone. I tried to transfer all of my data from my old iPhone to my new iPhone. The only thing that did not transfer is, of course, all of my music files (4,086 files in total). These files do exist on my computer as well, but the tags are wrong, and if I transferred them from my computer to the new iPhone, I would then have to spend 2 to 3 hours correcting the tags before they would be usable. Is there a way to transfer these files directly from my old iPhone and, if so, how?

I apologise if this comes across as hostile or unpleasant, I'm just trying to be as clear as possible because none of the searches I've done have turned up any information that is relevant to my current situation (I don't use iCloud and nor is any of my music purchased from the iTunes Store or Apple Music, etc).

iPhone XS/iOS 14.3

iPhone 6/iOS 12.3.1

MacOS Sierra/10.12.6 with iTunes 12.8.2.3