The key features of density-functional theory (DFT) within a minimalistic implementation of quantum electrodynamics are demonstrated, thus allowing to study elementary properties of quantum-electrodynamical density-functional theory (QEDFT). We primarily employ the quantum Rabi model that describes a two-level system coupled to a single photon mode and also discuss the Dicke model, where multiple two-level systems couple to the same photon mode. In these settings, the density variables of the system are the polarization and the displacement of the photon field. We give analytical expressions for the constrained-search functional and the exchange-correlation potential and compare them to established results from QEDFT. We further derive a form for the adiabatic connection that is almost explicit in the density variables, up to only a nonexplicit correlation term that gets bounded both analytically and numerically. This allows several key features of DFT to be studied without approximations.