In the context of first-passage percolation (FPP), we investigate the statistical properties of the selected link-times (SLTs) -the random link times comprising the optimal paths (or geodesics) connecting two given points. We focus on weakly disordered square lattices, whose geodesics are known to fall under the Kardar-Parisi-Zhang (KPZ) universality class. Our analysis reveals universal power-law decays with the end-to-end distance for both the average and standard deviation of the SLTs, along with an intricate pattern of long-range correlations, whose scaling exponents are directly linked to KPZ universality. Crucially, the SLT distributions for diagonal and axial paths exhibit significant differences, which we trace back to the distinct directed and undirected nature, respectively, of the underlying geodesics. Moreover, we demonstrate that the SLT distribution violates the conditions of the central limit theorem. Instead, SLT sums follow the Tracy-Widom distribution characteristic of the KPZ class, which we associate with evidence for the emergence of high-order long-range correlations in the ensemble.