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Recent studies of organic molecular solids have focused on their complex phase diagram and on light-induced phenomena, including a Mott insulating state, a spin liquid phase, and light-enhanced superconductivity. However, discrepancies between experiments and first-principles calculations for the κ-(BEDT-TTF)2X family hinder a comprehensive understanding of their properties. Here, we revisit the electronic structure of κ-(BEDT-TTF)2Cu2(CN)3 with a recently developed method for applying the Hubbard U potential on generalized orbital states, within the framework of density functional theory, to correct the orbital energy levels of the molecular solid. Our work focuses on the electronic structure of κ-(BEDT-TTF)2Cu2(CN)3, whose insulating state originates from an energy gap between the highest occupied and the lowest unoccupied molecular orbital states of the BEDT-TTF dimers, which constitute the periodic unit of the molecular solid. Our calculations provide results in alignment with experiments for band gaps, optical conductivities, and evolution of the metal-insulator transition as a function of pressure. Especially, the observed superconducting dome of κ-(BEDT-TTF)2Cu2(CN)3, which derives from the flat band state at the Fermi level, is qualitatively reproduced. Additionally, we construct a new low-energy lattice model based on our first-principles computed band structure that can be exploited to address many-body physics, such as quantum spin liquid states and double-holon dynamics. Our work can be extended to achieve deeper insight into the complex phase diagram and light-induced phenomena in the κ-(BEDT-TTF)2X family and other complex organic molecular solids.