Your questions are really too broad to answer in this forum. One of the strengths of Classical Thermodynamics is that it is "path independent". As your textbook stated, this means it is primarily focused on the starting and end points and not any intermediate states. This could be looked at as a weakness, since it doesn't help much distinguish between alternative paths, or it could be (and is) looked at as a strength since your answer/prediction isn't dependent on (doesn't rely on) the intermediate states (path). In other words, it gains predictive power by limiting itself to path independent states.
There's a lot you can know about a process, given only its initial and final conditions. One of the things this allowed was the development of Classical Thermodynamics BEFORE the discovery of the atomic nature of matter! Just think about that! Before we even knew about atoms, we could predict what the energy balance (not to mention entropy) was of a particular reaction!
Another part of your question states Thermodynamics "doesn't apply" to non-equilibrium systems. First, there's a whole sub-category of Thermodynamics which does deal with systems far from equilibrium, if you're interested. Second, think about what you're asking. You should know (if you think about it) that nothing, nothing is "at" equilibrium...yet. Give it another 10E15 years. So if you actually think that Thermodynamics can't be applied until the Universe comes to thermal (and energetic) equilibrium, you're being foolish. It does apply to systems "near" equilibrium – which is enormously, stupendously useful (if it didn't it would be virtually useless, imho).
The way I look at it you give up the details (the intermediate steps) because otherwise the results would depend on them and if they did then we'd have to know what those steps (states or mechanisms) are in order to apply Thermodynamics – and every time we were wrong about the intermediate steps (which is quite often, they're much harder to pin down because they are occurring on much shorter time scales and are (by definition) much less stable) we'd be wrong about the Thermodynamics – we exchange the intermediate details with more powerful predictions about results.
It's like you asking why the Ideal Gas Law only applies to point particles without volume or with no interactions between them. Short answer: it applies, but not perfectly, to real world gasses. Likewise, thermodynamics applies, but not perfectly, to real world systems near equilibrium.