  Robust cooling and nondestructive imaging are prerequisites for many emerging
applications of neutral atoms trapped in optical tweezers, such as their use in
quantum information science and analog quantum simulation. The tasks of cooling
and imaging can be challenged, however, by the presence of large trap-induced
shifts of their respective optical transitions. Here, we explore a system of
$^{39}$K atoms trapped in a near-detuned ($780$ nm) optical tweezer, which
leads to relatively minor differential (ground vs. excited state) Stark shifts.
We demonstrate that simple and robust loading, cooling, and imaging can be
achieved through a combined addressing of the D$_\textrm{1}$ and D$_\textrm{2}$
transitions. While imaging on the D$_\textrm{2}$ transition, we can
simultaneously apply $\Lambda$-enhanced gray molasses (GM) on the
D$_\textrm{1}$ transition, preserving low backgrounds for single-atom imaging
through spectral filtering. Using D$_\textrm{1}$ cooling during and after trap
loading, we demonstrate enhanced loading efficiencies as well as cooling to low
temperatures. These results suggest a simple and robust path for loading and
cooling large arrays of potassium atoms in optical tweezers through the use of
resource-efficient near-detuned optical tweezers and GM cooling.
