Your reasoning is correct for the case of semiconductor since effective mass of charge carriers m* is indeed inversely proportional to the curvature of the band. Thus, the higher is the dispersion, the higher is the curvature and the lowest is the mass so the highest is the mobility of charge carriers and the higher is the conductivity.
For metals the charge carriers are electrons characterized by a real mass of electron and does not depend on the dispersion. What is depending on the dispersion if the current density calculated as the integral or the sum of the first derivatives of Ek with respect to k. So the higher is the dispersion the highest are the values of the dEk/dk in the Brillouin zone and the highest is the current density.
CC
For metals the charge carriers are electrons characterized by a real mass of electron and does not depend on the dispersion. What is depending on the dispersion if the current density calculated as the integral or the sum of the first derivatives of Ek with respect to k. So the higher is the dispersion the highest are the values of the dEk/dk in the Brillouin zone and the highest is the current density.
CC