I've always wondered why we didn't have, at least, one scene with Beth earlier in the show, after her divorce from Frank, to introduce the character, specifically so that "Not A Day Goes By" doesn't come out of the blue in the divorce scene. Just because people divorce doesn't mean that they disappear from one another's stories, although, in this case, I believe the reason for that is that Beth is so devastated that she takes the son and leaves the city to return to her hometown.
I would love to see a post-divorce/pre-ballad Beth who has survived crossing paths with Frank in a way that we're not sure Charlie or Mary will. Perhaps she went back to performing or maybe she got remarried to someone who makes her happy or maybe something entirely different. I imagine that, no matter what, Beth would have a private sadness that her marriage to Frank failed and that she will always love him in some small way, even though she would only show that to the audience or, perhaps, Charlie or Mary, but never to Frank.