We study the impact of mass-transfer physics on the observable properties of binary black hole populations formed through isolated binary evolution. We investigate the impact of mass-accretion efficiency onto compact objects and common-envelope efficiency on the observed distributions of $chi_{eff}$, $M_{chirp}$ and $q$. We find that low common envelope efficiency translates to tighter orbits post 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-waves detectors as they form and merge at high redshifts ($zsim 2$), 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 $chi_{eff}$ 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 the local rate density ($zsimeq 0.01$) for the common envelope channel is in the range $sim 17-113~Gpc^{-3}yr^{-1}$ considering a range of $alpha_{CE} in [0.2,5.0]$ while for the stable mass transfer channel the rate density is $sim 25~Gpc^{-3}yr^{-1}$. 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 constrain the lower bound of branching fraction from other formation channels in the detected population to be $sim 0.2$. 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.
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