You are right for spontaneous emission, there we need to connect the system to something else, or to formulate a detailed-enough relativistic model with retarde

You are right for spontaneous emission, there we need to connect the system to something else, or to formulate a detailed-enough relativistic model with retarded EM fields that is able to lose energy (ordinary Hamiltonian of charged particle variables only cannot describe loss of energy to radiation). But for spontaneous absorption/emission, you are not right. Describing atoms by psi function is extremely successful, it is what Schroedinger did in his papers, what got him the right results for positions and intensities of spectral lines and what put his equation to the center of interest. It is true this successful description has some problems, such as the question of consistency of different world cuts (into system and environment), but the ordinary way to do the split for atom-radiation interaction (time-independent atom Hamiltonian and time-dependent external EM field) describes stimulated emission and absorption very well. Stimulated absorption/emission does require presence of external field, but this field can be taken into account in the usual psi function formulation - this is called semi-classical theory of radiation, it is what Schroedinger proposed in 1920's for simple atoms and is still heavily used for description of interaction of radiation with atoms and molecules.