  Unaccounted for systematics from foregrounds and instruments can severely
limit the sensitivity of current experiments from detecting redshifted 21~cm
signals from the Epoch of Reionization (EoR). Upcoming experiments are faced
with a challenge to deliver more collecting area per antenna element without
degrading the data with systematics. This paper and its companions show that
dishes are viable for achieving this balance using the Hydrogen Epoch of
Reionization Array (HERA) as an example. Here, we specifically identify
spectral systematics associated with the antenna power pattern as a significant
detriment to all EoR experiments which causes the already bright foreground
power to leak well beyond ideal limits and contaminate the otherwise clean EoR
signal modes. A primary source of this chromaticity is reflections in the
antenna-feed assembly and between structures in neighboring antennas. Using
precise foreground simulations taking wide-field effects into account, we
provide a framework to set cosmologically-motivated design specifications on
these reflections to prevent further EoR signal degradation. We show HERA will
not be impeded by such spectral systematics and demonstrate that even in a
conservative scenario that does not perform removal of foregrounds, HERA will
detect EoR signal in line-of-sight $k$-modes, $k_\parallel \gtrsim
0.2\,h$~Mpc$^{-1}$, with high significance. All baselines in a 19-element HERA
layout are capable of detecting EoR over a substantial observing window on the
sky.
