"What do you think is the problem that push hands is trying to solve, and that is mostly solved with a stinging slap too?"
It's a good question, especially the way that Chris couched it, because very little in the world is about what is and much is about what we think stuff is.
So here goes, Chris. Both push hands and sticky hands, IMO, are sensitivity drills. They are designed so that if you get good you can touch your opponent (not just in sparring, but a real bad guy) or get touched by him and know, in an instant, where every joint and bone of his body is, where they are moving towards and where they are about to move.
The next step, and it may take longer than an instant of contact, maybe a quarter of a second, you are drawn to where his structure can be manipulated, whether it is strong structure that can act as a lever arm against his base or weak structure that can be folded.
That, IMO is what the drill is designed to do. That is the application. The drill becomes a thing in itself very quickly. A game played by its own rules no longer connected with a separate purpose. So people play at maintaining structure moving and disrupting one another's structure. And that's a fine skill too, but a very easy skill to remove from context.
Within the game, a quick, stinging slap destroys the concentration, and that, in most cases, shatters the person's structure. Because in the game, structure comes from concentration. If you are doing it right, in real life, structure is a habit and a sudden sharp pain will move you towards habit. It is separate from concentration.
Which may be why judo players have better structure under stress than tai chi players (again, IME)--when your thoughts focus on tactics while people are trying to knock you down, structure must become a habit. When structure is the focus of your thoughts, always conscious, it never digs into the older parts of the brain. Just a theory.
A lot of training drills designed to handle critical fractions of a second get abstracted and become games. If you think about it, the benefits I describe here for push hands and sticky hands are the exact benefits I try to draw out with blindfolded in-fighting drills. A little more consciously, a little more context-aware... but the basic purpose is the same.