This is a fascinating thread. I had never thought about the question of monologues in musicals before. I watched the Richard Burton video and was mesmerized. However, I found myself wondering whether his monologue was meant to be sung instead of spoken, given that it was spoken to music. Also, I wonder whether the reason there are so few monologues in musicals is that a character's rumination usually takes place in song, even if the piece would be equally (or more) moving as a spoken monologue. The examples I can think of are "It's a Puzzlement," from The King and I, "Soliloquy," from Carousel, "I've Grown Accustomed to Her Face," from My Fair Lady, and Jean Valjean's soliloquy (whose name I've forgotten) from Les Miz, where he vows to leave JVJ behind.
Audrey, the Phantom Phanatic, who nonetheless would rather be Jean Valjean, who knew how to make lemonade out of lemons.