Hi.
While Apple may be trying to abandon iTunes the application itself is still available, is being updated, and is a much more powerful media manager than Apple Music. The only possible benefit I can find for Windows users who also subscribe to the Apple Music subscription service is the ability to access hi-res streams/downloads. Installing any of Apple Music/Devices/TV for Windows hobbles iTunes such that it can thereafter only manage audiobooks and podcasts. Device management duties are suppressed in favour of Apple Devices. I recommend removing all three of the new apps so you can use the program you are already familiar with and have invested time in.
Before we move on, it is possible to update composer details in Apple Music. Select some tracks then use the three dots menu or right-click and choose Properties. That said, I still advise reverting to iTunes.
Should you need to reinstall iTunes for Windows use this direct link from Apple:
To be clear iOS/iPadOS have an iTunes Store app that you can use to make purchases, but music synced to the device is sent to the Apple Music app whether your computer is running iTunes or Apple Music.
If your device won't show up in iTunes when connected see iOS device not showing in iTunes for Windows - Apple Community.
If you have trouble syncing your content to your device, or it won't play once it is there, see If you have music that won't play on your device - Apple Community. In some cases it seems iTunes or Apple Music can get confused about what is and isn't already copied over, and repeated syncing doesn't resolve things. Removing the music on the device and starting a fresh sync can clear this up. In the worst cases the device needs to be backed up, restored to factory settings, have the backup restored, and then synced with your media.
Given that you have iTunes Match you should really be connecting any device that supports it (not the iPod classic) to your iCloud Music Library so that changes you make on any device, plays, playlist updates, metadata changes, etc. sync seamlessly from one device to another through the cloud. See Use Sync Library with your Apple Music subscription - Apple Support. The same steps detailed there work for iTunes Match. For this to work correctly all of your songs should be matched, uploaded, or purchased. Use the Songs view and enable the iCloud Status column. If you have tracks marked as waiting, error, duplicate, ineligible, etc. then we can discuss how to resolve those.
tt2