A brief philosophical disclaimer: the way in which physics answers "why" questions like yours is with mathematical models. This is sometimes unsatisfying to those of us who are more comfortable with answers based either on logic or on our everyday knowledge and experience. Unfortunately, the laws of nature are not (as far as we know) logically necessary, and some of them, like the laws of quantum mechanics, are alien to our experience as well. The point is that the following explanation, which really just describes the physical law that makes atoms behave in this way, might not be the kind of "why" answer you are looking for, but it's the only kind physics knows how to give.
The reason atoms only absorb photons of the right energy is that (as we know from experiments) electrons obey the Schrodinger equation. For bound particles (i.e. particles trapped in some kind of potential, like the electrons that are trapped orbiting the atomic nucleus), the Schrodinger equation only allows discrete solutions, meaning that the particle can only occupy specific energy levels and is not allowed to exist in states of intermediate energy. This Hyperphysics discussion of the Schrodinger equation and the particle-in-a-box solution might be a good place to start building intuition about these concepts.
This means that an electron bound to an atomic nucleus can only change energy levels if it is given exactly the right amount of energy to transition to another state allowed by the Schrodinger equation. If a photon of a slightly lower or higher energy arrives, the atom will not absorb it because it can't do anything with that amount of energy - the electron is simply not allowed to have an energy that does not satisfy the Schrodinger equation. The "transient unstable orbit[s]" you imagine, while intuitively appealing, just aren't permitted by the laws of quantum mechanics that (according to all our observations) electrons obey.
Part of your concern seems to be that an electron in one energy level has some mysterious knowledge of the other energy levels it can theoretically occupy, and then decides based on that knowledge whether or not to absorb a photon. If that is a source of confusion, it might help to think of the "knowledge" as being contained in the whole system - the atom, photon, and the laws of quantum mechanics that they obey. If the photon has the right amount of energy, the two allowed electron states are coupled, and a transition can occur; if not, the coupling does not exist, and there is no transition. This idea of the knowledge of how to behave being contained in the whole system is really what we mean by a physical law, so perhaps it feels less puzzling than imagining that the electron alone does all the deciding.