Combining experimental data, numerical transport calculations, and theoretical analysis, we study the temperature-dependent resistivity of high-mobility two-dimensional (2D) Si MOSFETs to search for signatures of weak localization induced quantum corrections in the effective metallic regime above the critical density of the so-called two-dimensional metal-insulator transition (2D MIT). The goal is to look for the effect of logarithmic insulating localization correction to the metallic temperature dependence in the 2D conductivity so as to distinguish between the 2D MIT being a true quantum phase transition versus being a finite-temperature crossover. We use the Boltzmann theory of resistivity including the temperature-dependent screening effect on charged impurities in the system to fit the data. We analyze weak perpendicular field magnetoresistance data taken in the vicinity of the transition and show that they are consistent with weak localization behavior in the strongly disordered regime k(F)l greater than or similar to 1. Therefore, we supplement the Boltzmann transport theory with a logarithmic in temperature quantum weak localization correction and analyze the competition of the insulating temperature dependence of this correction with the metallic temperature dependence of the Boltzmann conductivity. Using this minimal theoretical model, we find that the logarithmic insulating correction is masked by the metallic temperature dependence of the Boltzmann resistivity and therefore the insulating ln T behavior may be apparent only at very low temperatures which are often beyond the range of temperatures accessible experimentally. Analyzing the low-T experimental Si MOSFET transport data, we identify signatures of the putative insulating behavior at low temperature and density in the effective metallic phase.