We present a model of the short gamma-ray burst (SGRB) population under a ‘quasi-universal jet’ scenario in which jets can differ somewhat in their on-axis peak prompt emission luminosity, L_c, but share a universal angular luminosity profile, ℓ(θ_v) = L(θ_v)/L_c, as a function of the viewing angle, θ_v. The model was fitted, through a Bayesian hierarchical approach inspired by gravitational wave (GW) population analyses, to three observed SGRB samples simultaneously: the Fermi/GBM sample of SGRBs with spectral information available in the catalogue (367 events); a flux-complete sample of 16 Swift/BAT SGRBs that are also detected by the GBM and have a measured redshift; and a sample of SGRBs with a binary neutron star (BNS) merger counterpart, which only includes GRB 170817A at present. Particular care was put into modelling selection effects. The resulting model, which reproduces the observations, favours a narrow jet ‘core’ with half-opening angle θ_c = 2.1_−1.4^+2.4 deg (uncertainties hereon refer to 90% credible intervals from our fiducial ‘full sample’ analysis) whose peak luminosity, as seen on-axis, is distributed as a power law, p(L_c) ∝ L_c^−A with A = 3.2_−0.4^+0.7, above a minimum isotropic-equivalent luminosity, L_c^⋆ = 5₋₂⁺¹¹ × 10⁵¹ erg s⁻¹. For viewing angles larger than θ_c, the luminosity profile scales as a single power law, l ∝ θ_v^−α_L with α_L = 4.7_−1.4^+1.2, with no evidence of a break, despite the model allowing for it. While the model implies an intrinsic ‘Yonetoku’ correlation between L and the peak photon energy, E_p, of the spectral energy distribution, its slope is somewhat shallower, E_p ∝ L^{0.4 ± 0.2}, than the apparent one, and the normalisation is offset towards larger E_p due to selection effects. The implied local rate density of SGRBs (regardless of the viewing angle) is between about one hundred up to several thousand events per cubic gigaparsec per year, in line with the BNS merger rate density inferred from GW observations. Based on the model, we predict 0.2 to 1.3 joint GW+SGRB detections per year by the advanced GW detector network and Fermi/GBM during the O4 observing run.