We study the impact of mass-transfer physics on the observable properties of binary black hole populations that formed through isolated binary evolution. We used the POSYDON framework to combine detailed MESA binary simulations with the COSMIC population synthesis tool to obtain an accurate estimate of merging binary black hole observables with a specific focus on the spins of the black holes. We investigate the impact of mass-accretion efficiency onto compact objects and common-envelope efficiency on the observed distributions of the effective inspiral spin parameter χeff, chirp mass Mchirp, and binary mass ratio q. We find that low common envelope efficiency translates to tighter orbits following the common envelope and therefore more tidally spun up second-born black holes. However, these systems have short merger timescales and are only marginally detectable by current gravitational-wave detectors as they form and merge at high redshifts (z ∼ 2), outside current detector horizons. Assuming Eddington-limited accretion efficiency and that the first-born black hole is formed with a negligible spin, we find that all non-zero χeff systems in the detectable population can come only from the common envelope channel as the stable mass-transfer channel cannot shrink the orbits enough for efficient tidal spin-up to take place. We find that the local rate density (z ≃ 0.01) for the common envelope channel is in the range of ∼17–113 Gpc−3 yr−1, considering a range of αCE ∈ [0.2, 5.0], while for the stable mass transfer channel the rate density is ∼25 Gpc−3 yr−1. The latter drops by two orders of magnitude if the mass accretion onto the black hole is not Eddington limited because conservative mass transfer does not shrink the orbit as efficiently as non-conservative mass transfer does. Finally, using GWTC-2 events, we constrained the lower bound of branching fraction from other formation channels in the detected population to be ∼0.2. Assuming all remaining events to be formed through either stable mass transfer or common envelope channels, we find moderate to strong evidence in favour of models with inefficient common envelopes.