It is shown that a broad class of cavity quantum electrodynamics (QED) problems - which consider the resonant propagation of a single photon interacting with quantum emitters (QEs), such as atoms, quantum dots, or vacancy centers - can be solved directly without application of the second quantization formalism. In the developed approach, the Hamiltonian is expressed through the ket-bra products of collective (photon+cavities+QEs) states. Consequently, the S matrix of input-output problems is determined exactly by the Mahaux-Weidenmüller formula, which dramatically simplifies the analysis of complex cavity QED systems. First, this approach is illustrated for the problem of propagation of a photon resonantly interacting with N two-level QEs arbitrary distributed inside the optical cavity. Solution of this problem manifests the effect of cumulative action of QEs previously known for special cases. Can a similar cumulative action of QEs enhance the inelastic resonant transmission of a single photon? We solve this problem for the case of an optical cavity having two modes resonantly coupled to electronic transitions of N three-level QEs. It is shown that the described structure is the simplest realistic structure which enables the down-conversion of the single-photon frequency with the amplitude approaching unity in the absence of the external driving field and sufficiently small cavity losses and QE dissipation. Overall, the simplicity and generality of the developed approach suggest a practical way to identify and describe new phenomena in cavity QED.