Tunes/Music maintains its own internal metadata database, a copy of metadata stored in audio files. Every time when you edit audio metadata in iTunes, it writes it to its own database. Then it syncs the database with the metadata embedded to audio files.
Unfortunately, Rating, Plays, and there are few other fields that are the exception. iTunes/Music does not write or embed the content of these fields into audio files.
You can import the content of Rating, Plays and any other iTunes/Music fields with the Edit / Load Tags / From Music menu command.
Import Audio Tracks from iTunes/Music
To make the command Edit / Load Tags / From Music work, you should import the files from iTunes/Music so that Tag Editor can search for audio tracks in iTunes by their paths.
Choose the File / Import from Music menu command. Alternatively, you can press ⌥⌘O or hit the link 'Import from Music' in the main window.
Import from Music/iTunes menu command
In the Import from Music window, select the playlist and press Continue. If you want to import the entire Music library, select the 'Music' playlist.
Import from Music/iTunes window
Show Rating and Plays columns
Rating and Plays columns are not visible by default, so you need to show them. This could be done both via main or context menu 'View / Column / Rating' and 'View / Column / Plays'.
Import Ratings and Plays
Select a few cells in the Rating and Plays columns. You can do that with the keyboard shortcuts or via the top or context menu 'Edit / Select Column'.
When selection is made, choose the menu Edit / Load Tags / From Music. Alternatively, you can do that with the Action menu as in the screenshot below.
Loading Ratings and Plays from Music/iTunes
When you finished, check the result of importing Plays and Ratings. Save the changes via ⌘+S or File / Save menu.
Ratings and Plays imported from Music/iTunes