Within the context of a general mode-coupling theory previously developed for the description of equilibrium fluctuations in simple classical fluids, we derive an expression for the effective longitudinal viscosity including couplings to currents so that certain key assumptions inherent in previous work on the glass transition can be analyzed critically. We show that the hopping kernels arise from an approximate resummation of the dominant class of diagrams resulting from bilinear modes. Although the resummed expression for the viscosity obtained in this paper is very similar to the result previously obtained from a kinetic theory approach, there are important differences, most notably in the form of the vertices connecting the hopping kernels and the lack of subtractions in the kernels representing the cage effect and activated hopping transport. Using the formalism developed presently, we show that the introduction of an initial~cutoff time in the hopping kernel to obtain a strong temperature dependence for the viscosity amounts to neglecting a class of processes that may be of comparable if not greater importance than the terms explicitly kept in previous work.