In 2018 the EDGES experiment claimed the first detection of the global cosmic 21 cm signal, which featured an absorption trough centered around z ∼ 17 with a depth of approximately −500 mK. This amplitude is deeper than the standard prediction (in which the radio background is determined by the cosmic microwave background) by a factor of two and potentially hints at the existence of a radio background excess. While this result was obtained by fitting the data with a phenomenological flattened-Gaussian shape for the cosmological signal, here we develop a physical model for the inhomogeneous radio background sourced by the first galaxies hosting population III stars. Star formation in these galaxies is quenched at lower redshifts due to various feedback mechanisms, so they serve as a natural candidate for the excess radio background indicated by EDGES without violating present-day measurements by ARCADE2. We forward-model the EDGES sky temperature data, jointly sampling our physical model for the cosmic signal, a foreground model, and residual calibration errors. We compared the Bayesian evidence obtained by varying the complexity and prior ranges for the systematics. We find that the data are best explained by a model with seven log-polynomial foreground terms and a component accounting for calibration residuals. Interestingly, the presence of a cosmic 21 cm signal with a non-standard depth is decisively disfavored. This result is contrary to previous EDGES analyses in the context of extra radio background models, thus serving as a caution against using a “pseudo-likelihood” built on a model (flattened Gaussian) that is different from the one being used for inference. We make our simulation code and associated emulator publicly available.