How can I easily migrate all my iTunes songs and artists to a new MacBook without a permanent external hard drive?

I've not peered into it deeply, but I imagine that system migration is primarily concerned with copying over the content of your user folder from one system drive to the other. It probably doesn't come with any deeper investigation of what dependencies there might be on additional storage locations.

To move the library easily from one system to another you need to copy both the library database, and the media folder, such that the media folder is at the same path on the new system, or start with the library in a portable shape (the media folder is in the same folder as the .musiclibrary package) that can automatically adjust when opened from a new drive/location. If there is content outside of the media folder this should be consolidated before the migration, or moved to exactly the same paths on the new system so it will be available and can be consolidated later.

The library export functions create text files that list the content of the library. Importing one of these into an empty library only works if every file listed is at the exact path specified, often not the case when the user name on a new machine might be different, or an external drive hasn't been moved over. The new library will, at best, have new date added values for all content. At worst playlists and content will be incomplete or missing because they won't import. When the library is moved correctly nothing is imported or exported, the library database is accessed on the new machine, and if the media is where it is supposed to be it all just works. If the media isn't in quite the right place it will be listed in the library, but won't play. There are ways to fix that, either by moving things around a little, tweaking a preference file, or using a script to effect repairs. We can cross that bridge if we get there.

Once the library is working on the new system, and presuming you have enough space on the internal drive(s), we can talk further about how to consolidate the media into the traditional path of ~/Music/Music/Media so that the library isn't dependent on the external HDD.

tt2

Your music isn’t stolen — your library folder structure wasn’t transferred properly. Moving only the files breaks Apple Music’s ability to track them.

Fix: Move the entire iTunes/Music library folder, then open Music.app while holding Option and choose that library. This restores all your music, artists, playlists, and artwork without a subscription. If your library is damaged, use PowerTunes to rebuild it cleanly.

As for what holding down the Option key is all about see Use multiple libraries in Music on Mac - Apple Support. Unlike most applications you cannot simply double click on a Music or iTunes library file and expect the app to open with that file, instead it opens with the last library it successfully opened. This might even be the copy in the trash if you tried deleting that and copying your old library into the ~/Music/Music folder. Holding the Option key down at launch time is what gives you the ability to switch between libraries, create a new one, or open a library at non-standard path, something you might need to do after moving the library to a new computer.

I see you've just posted while I was writing the above. To be clear I haven't yet asked you to use the hold-option-method to switch libraries. You only do that if you need to. When you make a new library it will be empty, apart from your unhidden iTunes purchase history. Your old library is likely still there if you swap back to it. If you are running Catalina or later then you should be using Music rather than iTunes. Music uses the .musiclibrary package to store the library. The library database is distinct from the media on your computer. The library is an index of all of the items added to it, along with any playlists you have created.

tt2

Without already knowing specific details of your system I'm trying to provide a more generalised set of instructions that you or anyone else could adapt to their own circumstances. If you want me to give step by step advice then I will need details.

1. On the computer that has your working library now what is the path to the active .musiclibrary package that you are using. This is normally ~/Music/Music/Music Library.musiclibrary. The package gets updated every time you do anything will the library so you should be able to check which is it is. If you are not sure search the entire computer for all .musiclibrary files and compare dates.
2. With the library open look under Music > Settings (or Preferences) > Files and note the location of the Music Media folder.
3. Are you certain that everything connected to the library is in the location you found for 2?
4. How big is your library? Use View > Show Status Bar if not already present.
5. How big is your external drive? How much free space does it have?
6. How big is the internal drive on the new computer? How much free space does it have?

tt2

See also Move your iTunes/Music library to a new computer - Apple Community.

tt2

Follow up questions.

1. What version of macOS is on the 2010 Mac Pro? Presumably this is running iTunes with an old version of your library.
2. What version of macOS in on the 2018 MacBook Pro? If running Catalina or later you should be using Music on it, and in general you don't now want to open older .itl files on it, though there could potentially be specific circumstances when you want to do this, if for example you know that one of those libraries held more content than is currently listed in Music.
3. What version of macOS is on the 2020 MacBook Air?

I understood that you wanted to copy the library from the 2018 model to the 2020 model? Why did you make any changes to the old library? A key part of cloning a library to a new system is not to make any changes to the old one.

tt2