Q: iTunes suddenly showing new install screen?

This is the screen you see when you use a brand new iTunes library.  Are you sure you did not log into a different user account on the Mac by mistake?  A new user account would have a new iTunes library.

This can also happen if your iTunes library becomes corrupted.  iTunes opens to a new iTunes library.

Quit iTunes, if running.  In a Finder window, open it to

[startup disk]/Users/[user account name]/Music/iTunes/

Do you see a file called iTunes Library.itl there?  That is the default location for your iTunes library database file.  In case iTunes got set to open a new library, hold down Option key and launch iTunes; you are doing this procedure

Open a different iTunes Library file or create a new one - Apple Support

Navigate to and select that iTunes Library.itl file you saw in Finder window.  If iTunes is able to open it, that should be your usual iTunes library.

Even if the iTune library becomes corrupted, the media files (such as song files) are probably fine.  The default location for storing the media files is in that Music/iTunes folder, in a folder called iTunes Media.  They are organized into sub-folders by media type (Music for songs), then by artist, album, and song name (the actual files).  You should be able to use a Finder window, and find all of you song files (including any that are not in your backup).

There may also be a folder in the Music/iTunes folder called Previous iTunes Libraries.  I think iTunes makes a backup of your existing iTunes library when you update to a major new release of iTunes.  Apple has this document, which describes how to use one of the previous library files

If you don't see your content after you update iTunes - Apple Support

Your situation is not related to doing an iTunes update, but you can still use the latest "previous" iTunes library file.  Unfortunately, it may be a few months old.  I just checked, and the most recent iTunes library file I see in that folder is from July 2015.  But if I had to use it because my iTunes library became corrupted, I would get back things like playlists, date added, ratings, and play count (up to the date of the file).  I'd have to manually add back all songs (and other media) added after that date (which should be in the iTunes Media folder)

Actually, if you use Time Machine to back up your data, the BETTER place find the latest (uncorrupted) version of your iTunes library is in your Time Machine archive.  Remove (or move) the current corrupted iTunes library.itl file.  Go "back in time" to a state just before this problem existed, and restore it from that Time Machine state.  Try to use Option-launch of iTunes to open the restored library file.

Time Machine backs up changes every hour, so you can get very close to the point when the corruption occurred.

iTunes Music is the old folder structure used by iTunes.  When iTunes switched to the new iTunes Media folder structure, iTunes should have asked if you want to keep using the iTunes Music structure or have iTunes reorganize your media files into the iTunes Media structure.  But it should not cause a problem either way.

If you are using the iTunes Music structure, you should see sub-folders with artist names directly in the iTunes Music folder.  You may also see sub-folders with other media category names, such as Movies or Podcasts.

Incidentally, iTunes Music appears to have only artist-name folders, and the recent additions had only track titles.

That is odd.  Song files would normally not appear directly in the iTunes Music folder; they should be in a folder with artist name, and then in folder with album name.  Are these "recent additions" purchased songs from iTunes Store, or song files you added manually?

However, there are definitely some recent music recordings missing

If you know the name (or partial name) of one of those songs, open a Finder window, and using its Search field, look for any files with the song name.  This is just to check if the song files got stored somewhere else.

When an iTunes library becomes corrupted, is it always the iTunes.library.itl file that's at fault?

That file IS your iTunes library.

So, I guess you don't keep a Time Machine backup of your system and user data?  It comes in handy in situations like your iTunes library becoming corrupted.  You would be able to restore it to within one hour of when the corruption occurred.

But they still weren’t coming up in iTunes itself.

iTunes does not necessarily know about everything that's in the iTunes Music folder.  For example, if you copy a song file into the iTunes Music folder using Finder (not saying that you did that), the iTunes library will not know about it.  So, adding them again to your iTunes library is the right move.

In iTunes, look at iTunes preferences Advanced pane.  Without changing anything, what does it currently say for iTunes Media folder location (you can replace your actual user account name with "My User Name")?  Also, are the TWO checkboxes for Keep iTunes Media folder organized AND Copy files to iTunes Media folder when adding to library currently checked or not checked?