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Star formation in galaxies is governed by the amount of molecular gas and the efficiency that gas is converted into stars. However, assessing the amount of molecular gas relies on the CO-to-H2 conversion factor (α_CO), which is known to vary with molecular gas conditions like density, temperature, and dynamical state – the same conditions that also alter star formation efficiency. The variation of α_CO, particularly in galaxy centers where α_CO can drop by nearly an order of magnitude, thus causes major uncertainties in current molecular gas and star formation efficiency measurements. Using ALMA observations of multiple 12CO, 13CO, and C18O lines in several barred galaxy centers, we found that α_CO is primarily driven by CO opacity changes and therefore shows strong correlations with observables like velocity dispersion and 12CO/13CO line ratio. Motivated by these results, we have established a new α_CO prescription which accounts for CO emissivity variations and verified it across a set of nearby galaxies with independent α_CO measurements from dust. Applying our new prescription to 65 galaxies from the PHANGS-ALMA survey, we found an overall 3x higher star formation efficiency in barred than non-barred galaxy centers, which is an unprecedented trend that has been obscured by past α_CO prescriptions. Our results suggest that the high star formation rates observed in barred galaxy centers is mainly due to an enhanced star formation efficiency, rather than a substantially increased amount of molecular gas.