  The LIGO and Virgo gravitational-wave detectors have uncovered binary black
hole systems with definitively nonzero spins, as well as systems with
significant spin residing in the more massive black hole of the pair. We
investigate the ability of isolated binary evolution in forming such highly
spinning, asymmetric-mass systems through both accretion onto the first-born
black hole and tidal spin-up of the second-born black hole using a rapid
population synthesis approach with detailed considerations of spin-up through
tidal interactions. Even with the most optimistic assumptions regarding the
efficiency at which an accreting star receives material from a donor, we find
that it is difficult to form systems with significant mass asymmetry and
moderate or high spins in the primary black hole component. Assuming efficient
angular momentum transport within massive stars and Eddington-limited accretion
onto black holes, we find that $< 1.5\%$ of systems in the underlying binary
black hole population have a primary black hole spin greater than 0.2 and a
mass asymmetry of greater than 2:1 in our most optimistic models, with most
models finding that this criteria is only met in $\sim 0.01\%$ of systems. The
production of systems with significant mass asymmetries and spin in the primary
black hole component is thus an unlikely byproduct of isolated evolution unless
highly super-Eddington accretion is invoked or angular momentum transport in
massive stars is less efficient than typically assumed.
