All lipid membranes have inherent morphological preferences and resist deformation. Yet adaptations in membrane shape can and do occur at multiple length scales. While this plasticity is crucial for cellular physiology, the factors controlling the morphological energetics of lipid bilayers and the dominant mechanisms of membrane remodeling remain unclear. An ongoing debate regarding the universality of the stiffening effect of cholesterol underscores the challenges facing this field particularly for lipid mixtures, both experimentally and theoretically. On the computational side, we have argued that enhanced-sampling molecular dynamics simulations are uniquely suited for quantification of membrane conformational energetics, not only because they minimize a-priori assumptions, but also because they permit analysis of bilayers in deformed states. To showcase this approach, we examine reported inconsistencies between alternative experimental measurements of bending moduli for cholesterol-enriched membranes. Specifically, we compute bending free-energy landscapes for multiple bilayers of sizes up to ~2,000 lipids, using steady-state all-atom simulations of both unperturbed and deformed morphologies, totaling over 100 microseconds of sampling. This enhanced simulation approach enables direct derivation of bending moduli in different contexts, while dissecting the contributing factors and underlying mechanisms, be it lipid tilt, changes in chain flexibility, or hydrophobic hydration. Our results are in excellent agreement with giant-vesicle measurements, confirming that cholesterol effects are lipid-specific and explaining why certain experiments probing the nanometer scale diverge. In summary, we demonstrate that quantitative structure-mechanics relationships can now be established for mixed lipid membranes, paving the way to addressing open central questions in cell membrane mechanics.