  We conduct binary population synthesis to investigate the formation of
wind-fed high-mass X-ray binaries containing black holes (BH-HMXBs). We evolve
multiple populations of high-mass binary stars and consider BH-HMXB formation
rates, masses, spins and separations. We find that systems similar to Cygnus
X-1 likely form after stable Case A mass transfer (MT) from the main sequence
progenitors of black holes, provided such MT is characterised by low accretion
efficiency, $\beta \lesssim 0.1$, with modest orbital angular momentum losses
from the non-accreted material. Additionally, efficient BH-HMXB formation
relies on a new simple treatment for Case A MT that allows donors to retain
larger core masses compared to traditional rapid population-synthesis
assumptions. At solar metallicity, our Preferred model yields $\mathcal{O}(1)$
observable BH-HMXBs in the Galaxy today, consistent with observations. In this
simulation, $8\%$ of BH-HMXBs go on to merge as binary black holes or neutron
star-black hole binaries within a Hubble time; however, none of the merging
binaries have BH-HMXB progenitors with properties similar to Cygnus X-1. With
our preferred settings for core mass growth, mass transfer efficiency and
angular momentum loss, accounting for an evolving metallicity, and integrating
over the metallicity-specific star formation history of the Universe, we find
that BH-HMXBs may have contributed $\approx2$--$5$ BBH merger signals to
detections reported in the third gravitational-wave transient catalogue of the
LIGO-Virgo-KAGRA Collaboration. We also suggest MT efficiency should be higher
during stable Case B MT than during Case A MT.
