I would say this differently: The slinky is a helical spring suspended from one end, so the tension force at any point along the spring is exactly what is necessary to support that part of the spring which is below that point.
In other words, it is not a coincidence that the force exactly balances gravity initially; it is the natural position of the spring, having the lowest potential energy.
It's fascinating to consider what would happen if a slinky were extended between two points along a horizontal friction-free surface, like an ice rink, and then one end of the slinky is released. My guess is that we would see a similar behavior, with the loose end collapsing toward the other end and a transition between collapsed coils and coils which are undisturbed from their initial positions.
Differently from the vertical case, the horizontal slinky would initially exhibit identical spacing between coils along its entire length.
We need somebody with a slinky, a high-speed camera, and an ice rink.
Interesting that you used that example. I had lunch with friend today. One of the "practically" smartest people one could meet in life. Engineer by training, brilliant at math but had a knack for communication and realized he could make more money in sales than in engineering. Long story short, he ended up buying one of the companies he worked for. It was a steel fabricator. The company did a lot tension steel.
Much of what you wrote about he was working out quickly in his head.
We started talking about grav forces and then we got talking about the exact horizontal example you mentioned above, "friction" free surface--ice--and restraining one end of the slinky while releasing the other.
You two are so close in what you each said. Since he had such a practical background of manufacturing tension steel, he really keyed in on the tensional relationship between each coil "loop" and force necessary to keep apart from the adjoining loop. This will remain same until it is acted upon.
He also brought up the question, if the slinky is stretched three stories and someone grabs a hold of it at some intermediate point, say somewhere in the second story, nothing should change in the tensional values below, even if the slinky is released above.
To bad science wasn't taught by guys like you in high school.