Aims. In order to understand the observed molecular diversity in high-mass star-forming regions, we have to determine the underlying physical and chemical structure of those regions at high angular resolution and over a range of evolutionary stages. Methods. We present a detailed observational and modeling study of the hot core VLA 3 in the high-mass star-forming region AFGL 2591, which is a target region of the NOrthern Extended Millimeter Array (NOEMA) large program CORE. Using NOEMA observations at 1.37 mm with an angular resolution of ~0″. 42 (1400 au at 3.33 kpc), we derived the physical and chemical structure of the source. We modeled the observed molecular abundances with the chemical evolution code MUSCLE (MUlti Stage ChemicaL codE). Results. With the kinetic temperature tracers CH₃CN and H₂CO we observe a temperature distribution with a power-law index of q = 0.41 ± 0.08. Using the visibilities of the continuum emission we derive a density structure with a power-law index of p = 1.7 ± 0.1. The hot core spectra reveal high molecular abundances and a rich diversity in complex molecules. The majority of the molecules have an asymmetric spatial distribution around the forming protostar(s), which indicates a complex physical structure on scales <1400 au. Using MUSCLE, we are able to explain the observed molecular abundance of 10 out of 14 modeled species at an estimated hot core chemical age of ~21 100 yr. In contrast to the observational analysis, our chemical modeling predicts a lower density power-law index of p < 1.4. Reasons for this discrepancy are discussed. Conclusions. Combining high spatial resolution observations with detailed chemical modeling allows us to derive a concise picture of the physical and chemical structure of the famous AFGL 2591 hot core. The next steps are to conduct a similar analysis for the whole CORE sample, and then use this analysis to constrain the chemical diversity in high-mass star formation to a much greater depth.