I'm gonna side with Hal Prince on this one... giving it to Beth, while in theory a great idea, is actually dramatically unsatisfying. She is never a character we get to know well, and when the song comes we've only just met her. What we need at that moment is more depth and emotional nuance from Frank, and that song gave him some of that. Giving it to Beth takes an important song and color of his character away from him and gives it to a character we don't know -- not to mention that she then really just has an anthem saying she will always love this man who cheated on her and broke up her family. It's only intellectually satisfying a little when, in act two, we realize she was singing back to him the vows he made on their wedding night. I don't think robbing Peter to pay Paul in this sense is worth it what you get from it.
Also the Act Two version of NADGB already has its own new element with Mary joining in singing it to/about Frank while Frank and Beth sing it to each other. That's enough for the audience.