I'm not sure if the Aeon article explains what Everett really meant, at least I still have questions. Suppose we're eternally branching, as in the example about the up/down spin. If there's no way to visit other branches, how will we ever confirm this? Second, what is the benefit of the branching interpretation? It seems to simply be a way to think about probabilities?
2. The benefit is that it's a very direct translation of the math into concepts. If you actually build the mathematical model, the "other worlds" are right there in the model, as the other half of a superposition.
It doesn't go away. You just can't get to it. Nor can you ignore it, because it's sitting there as a term in the equation, persisting as you evolve it over time. It just becomes smaller and smaller without ever becoming zero. That's why you observe a classical universe even though it's really quantum: the odds of the other parts of a superposition having an effect are on the order of Avogadro's-number-to-1 against.
Copenhagen works by just arbitrarily defining that number as zero, which is easy to work with but conceptually incomplete. MWI keeps it there in another "world", which is more in keeping with the math but unedifying because it sounds so unfalsifiable.
Hope that helps. (Which is always code for "... but it probably doesn't", I'm afraid.)