Context. In the first two papers of this publication series, we present a comprehensive library of synthetic Event Horizon Telescope (EHT) observations and used this library to train and validate Bayesian neural networks for the parameter inference of accreting supermassive black hole systems. The considered models are ray-traced general relativistic magnetohydrodynamic (GRMHD) simulations of Sgr A* and M87*. Aims. In this work, we infer the best-fitting accretion and black hole parameters from 2017 EHT data and predict improvements that will come with future upgrades of the array. Methods. Compared to previous EHT analyses, we considered a substantially larger synthetic data library and the most complete set of information from the observational data. We made use of the Bayesian nature of the trained neural networks and apply bootstrapping of known systematics in the observational data to obtain parameter posteriors. Results. Within a wide GRMHD parameter space, we find M87* to be best described by a spin between 0.5 and 0.94 with a retrograde MAD accretion flow and strong synchrotron emission from the jet. Sgr A* has a high spin of ∼0.8–0.9 and a prograde accretion flow beyond the standard MAD/SANE models with a comparatively weak jet emission, seen at a ∼ 20°–40° inclination and ∼106°–137° position angle. While previous EHT analyses could rule out specific regions in the model parameter space considered here, we are able to obtain narrow parameter posteriors with our ZINGULARITY framework without being impacted by the unknown foreground Faraday screens and data calibration biases. We further demonstrate that the Africa Millimeter Telescope extension to the EHT will reduce parameter inference errors by a factor of three for non-Kerr models, enabling more robust tests of general relativity. Conclusions. Our results agree with multiwavelength constraints from the literature. It will be instructive to produce new GRMHD models with the inferred interpolated parameters for in depth model-data comparisons and to study their accretion rate plus jet power.