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.
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