Our speaker is an outside observer, through and through. For a poem with such emotional baggage, though, it seems like an odd choice.
But think about it this way: since we're coming at this from an outside perspective, we as readers really get to do some serious spying. If it were told from, say, the grandmother's point of view, we would have a pretty limited idea of what's going on in the poem. And if it were told from the child's point of view, it would probably be even more limited, at least vocabulary-wise.
This way, we get a big-picture sense of the scene. Plus, we get to preserve the mystery. We don't have to hear why the grandmother's sad, or what this little girl's thinking that has her seeing tears on the tea kettle. Sure, we get some cameo lines from the grandmother, the stove, and the almanac, but we never dive into their consciousnesses for very long. Nothing much is revealed. Bishop would much rather let you revel in the strange and eerie atmosphere of the poem than tell you straight up why it's so. How boring would that be?