Take, for example, Facebook Top News. The benefit isn't that it's not real time, it's just that the algorithm for deciding what's top news isn't driven primarily by the post timestamp. In the future defaulting to Top News makes sense, but there's no reason that feed shouldn't be real-time. If a close friend gets engaged, I would want a notification about that as soon as possible. But I want Facebook to be smart about when it interrupts me.
The same is true for Twitter's weekly digest. I'd love to have the tweets Twitter picks out in real-time rather than only at the end of the week. I may not want to be notified in real time, but I wouldn't mind if they went in a special list I could browse every day. Basically whatever metrics Twitter uses to decide what are the best tweets for me, I want those applied is real-time, and when a story meets the threshold, let me know.