How to copy iTunes music to transfer to new computer?

You probably can't open the iTunes application in Catalina. However, what you want is the data stored in the iTunes Library in your user account.

I would suggest making (at least) two copies of this - one to save away in a safe place, another for use with Catalina. Follow the directions in the following Apple Support article to open the latter copy with Catalina and the Music app. As to merging things into one Music or TV library, I'm not sure what the steps are for that. But at least this should let you migrate your content.

https://support.apple.com/en-us/HT210230

Another possibility is to go into the iTunes / iTunes Music folder from the old computer, and retrieve individual .m4a, .mp3, etc., files. I don't have Catalina, so I don't know if you could drag the top-level folder onto the Catalina Music app/window and have it automatically walk the directory tree, adding everything that it finds to the current Catalina library.

The drag-and-drop? The original sound files would be in the backed-up iTunes folder, nested in individual folders (typically artist/album).

E.g., iTunes/iTunes Music/B. B. King/King Of The Blues/01\ (You've\ Become\ A)\ Habit\ To\ Me.m4a

I'm very surprised that your new machne got only the titles and the artwork ... unless, perhaps, it copied them, and imported the actual music "by reference" to the external hard drive (which you no longer have attached?!?). iTunes had an Advanced preference for whether to copy imported media files into its library; maybe Music has something similar, and it was on the "don't copy" setting.