Current searches for gravitational waves from compact-binary objects are primarily designed to detect the dominant gravitational-wave mode and assume that the binary components have spins which are aligned with the orbital angular momentum. These choices lead to observational biases in the observed distribution of sources. Sources with significant spin-orbit precession or unequal-mass-ratios, which have non-negligible contributions from subdominant gravitational-wave modes, may be missed; in particular, this may significantly suppress or bias the observed neutron star-black hole (NSBH) population. We simulate a fiducial population of NSBH mergers and determine the impact of using searches that only account for the dominant-mode and aligned spin. We compare the impact for the Advanced LIGO design, A+, LIGO Voyager, and Cosmic Explorer observatories. We find that for a fiducial population where the spin distribution is isotropic in orientation and uniform in magnitude, we will miss ∼25% of sources with mass-ratio q>6 and up to ∼60% of highly precessing sources (χp>0.5), after accounting for the approximate increase in background. In practice, the true observational bias can be even larger due to strict signal-consistency tests applied in searches. The observation of low spin, unequal-mass-ratio sources by Advanced LIGO design and Advanced Virgo may in part be due to these selection effects. The development of a search sensitive to high mass-ratio, precessing sources may allow the detection of new binaries whose spin properties would provide key insights into the formation and astrophysics of compact objects.