The summation method for the calculation of reactor $\bar{\nu}_e$ fluxes and spectra is methodically revised and improved. For the first time, a complete uncertainty budget accounting for all known effects likely to impact these calculations is proposed. Uncertainties of a few percents at low energies and ranging up to 20% at high energies are obtained on the calculation of a typical reactor $\bar{\nu}_e$ spectrum. Although huge improvements have been achieved over the past decade, the quality and incompleteness of the present day evaluated nuclear decay data still limit the accuracy of the calculations and therefore dominate by far these uncertainties. Pushing the $\beta$-decay modeling of the thousands of branches making a reactor $\bar{\nu}_e$ spectrum to a high level of details comparatively brings modest changes. In particular, including nuclear structure calculations in the evaluation of the non-unique forbidden transitions gives a smaller impact than anticipated in past studies. Finally, this new modeling is challenged against state-of-the-art predictions and measurements. While a good agreement is observed with the most recent Inverse Beta Decay measurements of reactor $\bar{\nu}_e$ fluxes and spectra, it is unable to properly describe the reference aggregate $\beta$ spectra measured at the Institut Laue-Langevin High-Flux reactor in the 80s. This result adds to recent suspicions $\beta$ the reliability of these data and preferentially points toward a misprediction of the $^{235}$U $\bar{\nu}_e$ spectrum.