  Charge order--ubiquitous among correlated materials--is customarily described
purely as an instability of the electronic structure. However, the resulting
theoretical predictions often do not match high-resolution experimental data. A
pertinent case is $1T$-VSe$_2$, whose single-band Fermi surface and
weak-coupling nature make it qualitatively similar to the Peierls model
underlying the traditional approach. Despite this, its Fermi surface is poorly
nested, the thermal evolution of its charge density wave (CDW) ordering vectors
displays an unexpected jump, and the CDW gap itself evades detection in direct
probes of the electronic structure. We demonstrate that the thermal variation
of the CDW vectors is naturally reproduced by the electronic susceptibility
when incorporating a structured, momentum-dependent electron-phonon coupling,
while the evasive CDW gap presents itself as a localized suppression of
spectral weight centered above the Fermi level. Our results showcase the
general utility of incorporating a structured coupling in the description of
charge ordered materials, including those that appear unconventional.
