© 2018 American Physical Society. A key interest in geomorphology is to predict how the shear stress τ exerted by a turbulent flow of air or liquid onto an erodible sediment bed affects the transport load Mg (i.e., the submerged weight of transported nonsuspended sediment per unit area) and its average velocity when exceeding the sediment transport threshold τt. Most transport rate predictions in the literature are based on the scaling Mg-τ-τt, the physical origin of which, however, has remained controversial. Here we test the universality and study the origin of this scaling law using particle-scale simulations of nonsuspended sediment transport driven by a large range of Newtonian fluids. We find that the scaling coefficient is a universal approximate constant and can be understood as an inverse granular friction coefficient (i.e., the ratio between granular shear stress and normal-bed pressure) evaluated at the base of the transport layer (i.e., the effective elevation of energetic particle-bed rebounds). Usually, the granular flow at this base is gaslike and rapidly turns into the solidlike granular bed underneath: a liquidlike regime does not necessarily exist, which is accentuated by a nonlocal granular flow rheology in both the transport layer and bed. Hence, this transition fundamentally differs from the solid-liquid transition (i.e., yielding) in dense granular flows even though both transitions are described by a friction law. Combining this result with recent insights into the nature of τt, we conclude that the transport load scaling is a signature of a steady rebound state and unrelated to entrainment of bed sediment.