I ran across Mildred Pierce in the bookstore today, and out of curiosity, I read the ending. It's a very good ending. But I still think the murder story works fine, too.
Honestly, I don't see any reason why they couldn't have used the book's ending in the original film, even in Production Code days. It's not much more salacious than what they did do. With the exception of the Veda-being-naked part, and that's easily solved -- just throw a robe on her. But they don't have the storyline about her becoming an opera singer, and without it, Cain's ending doesn't work. I wonder if they just decided that a murder story was easier and simpler to portray than the opera storyline.