This is a tough one. The field is too narrow. There are musicals that are basically filmed theatre (Sunday in the Park with George, Pirates of Penzance); musicals that are based on Broadway (Fiddler on the Roof, Funny Girl); musicals that are based on other works of art (Jesus Christ Superstar, My Fair Lady). But all of the great musicals are disqualified.
What do we have left? A lot of Disney:
Mary Poppins, which is great and has a lot of themes to it. The songs are sometimes counterpoint to the script, pointing out hypocrisies.
The Lion King, which is high drama.
Mulan, which is Disney's feminist manifesto, and hilarious whether in the songs or out.
Beauty and the Beast, which is vaguely taken from a fairy tale, but it's a great musical, with some very good lines.
Oh, and Once. Which is a great romance with or without the songs.
All of these films would be much weaker without the music, but they would stand, even so.