I think one way of answering this question is to look at the story as not something that is based in one contextual, historical period alone, but as offering us an archetype. The archetype of a journey is one that occurs many, many times in a variety of forms, as one central character tries to complete a journey, but encounters many barriers and obstacles on the way. You only have to consider characters such as Odysseus in the Odyssey to think of an excellent example. In the same way, Phoenix Jackson is a character with a definite goal, but she has to encounter and overcome a variety of obstacles on the path to her goal.
If we consider this archetypal reading, we can see that the themes presented in this excellent short story are truly timeless. All of us, at one stage or another, have to engage on some kind of journey or a quest: it may be to complete our studies or to get to a particular position in a job, but the archetype of a journey is one that is truly timeless and that we can all connect to. Often, as in the case of Phoenix Jackson, we set out on our own "worn path" for love, as she does for her grandson. The love that we have for another person often gives us the strength and endurance that we need to overcome the barriers on our "worn path," as love gives us strength.