By "locking in" the setting of the novel, we are saying "how is it necessary to the story that this action take place in the time and place that it does?" Clearly we could nudge any story in time and place and it would, hypothetically, still work, as long as the essential elements are intact. Thus we should look for elements that are unique to late 1800s New England, and understand how they inform the story.
The most significant element, which is alluded to but not focused on in the story, is the Second Industrial Revolution. This saw cities consolidating their power and influence even further; farms and the countryside were, in essence, becoming archaic. Considered from the perspective of the story's mood and message, this makes sense; Starkfield has seen better times, and piece by piece we get the picture that it is largely because the modern world is passing by. Ethan mentions that the trains don't stop in Starkfield, and Starkfield's winters are noted for their harshness. Starkfield is, in essence, withering, while other people and places blossom. We have a sense that life is better elsewhere, but these happier settings are just out of view.
Wharton was also specifically interested in portraying the type of New England that the novel conveys; one that is beautiful, but bleak, somber and somewhat anachronistic - she called it a "decaying rural existence". Clearly the reason that this wouldn't work for a comparable Southern or Western setting is the absence of the kind of winters necessary to frame the story in the proper melancholy light.
There are other minor considerations we can make as well; for example, Zenobia would have had better access to medical care in a more modern world, and in an older one the medicines she was constantly ordering may not have been available at all, hastening her death.