I'm not entirely sure how we answer this as an historical question, especially with a good, sourced, answer. The linked thread is pretty much a wide-eyed recounting of real work as ghost stories, and covers the not-exactly-specific location "the woods."

I suppose the thing I would address is that there are almost no old growth forests left in the United States, and no matter how deep in the woods you are, or how far from human contact it feels, chances are that people have been there before, felling, building, and moving on. Why are there staircases in the woods? Because they weren't always the woods, and somebody has been there in the last 200 years cutting them down. A well built set of stairs is liable to be one of the stronger parts of a structure, and one of the last to collapse or disintegrate away. Buildings would be built more frequently near areas of easier access (the same places people can reach to get lost), and their old roads would run along the same routes that make great hiking paths, and the very structures themselves are attractive to hikers and explorers (making them the first places SAR will search). It is entirely reasonable that a SAR operator encounters these things with some regularity, as they are clustered along the most heavily searched routes in the areas people are most likely to get lost.

Just to provide some context, I've done wilderness search and rescue myself, and many of the symptoms, hallucinations, and weird behaviors that poster describes are the sorts of things that are known to happen with people who have been out in the wilderness subjected to dehydration, hunger, pain, and hypothermia for a couple days. Shock, blood loss, infection, and cold all can dramatically warp the human mind, and we trained specifically for handling those kinds of behaviors - one oughtn't be surprised to encounter them in that setting.

I fully sympathize with the poster of that thread in regards to the cases that they couldn't resolve, and the sound of the bereaved mother... those sorts of things will eat at you, absolutely. They don't need any embellishment to strike a chord. Using all that to try and create some forest-service ghost conspiracy just feels disrespectful to the victims though, and if I were one of his coworkers, I too might tell him to shut up about the stairs and stop inventing wild internet tales so we could focus on the important work of saving lives.

In short, why are there stairs? Because there used to be buildings. The woods haven't always been the woods.

A couple of sources:

Information on forests, and how few spots there are that haven't been logged at some point in recent history (and why, therefore, one might find buildings out there).

A quick page on Search and Rescue Techniques, including mention and definition of "Likely Spots" in initial search, which would include old structures (and why, therefore, a SAR operator would encounter these buildings frequently).