  Proximity zones of high-redshift quasars are unique probes of supermassive
black hole formation, but simultaneously explaining proximity zone sizes and
black hole masses has proved to be challenging. We study the robustness of some
of the assumptions that are usually made to infer quasar lifetimes from
proximity zone sizes. We show that small proximity zones can be readily
explained by quasars that vary in brightness with a short duty cycle of
$f_\mathrm{duty}\sim 0.1$ and short bright periods of $t_\mathrm{on}\sim 10^4$
yr, even for long lifetimes. We further show that reconciling this with black
hole mass estimates requires the black hole to continue to grow and accrete
during its obscured phase. The consequent obscured fractions of $\gtrsim$ 0.7
or higher are consistent with low-redshift measurements and models of black
hole accretion. Such short duty cycles and long obscured phases are also
consistent with observations of large proximity zones, thus providing a simple,
unified model for proximity zones of all sizes. The large dynamic range of our
simulation, and its calibration to the Lyman-$\alpha$ forest, allows us to
investigate the influence of the large-scale topology of reionization and the
quasar's host halo mass on proximity zones. We find that incomplete
reionization can impede the growth of proximity zones and make them smaller up
to 30%, but the quasar host halo mass only affects proximity zones weakly and
indirectly. Our work suggests that high-redshift proximity zones can be an
effective tool to study quasar variability and black hole growth.
