In spite of notable advances in the description of river morphodynamics, the long-term dynamics of meandering rivers is still an open question, in particular, regarding the existence of a possible statistical steady state and its scaling properties induced by the competing action of cutoffs and reach elongation. By means of extensive numerical simulations, using three fluid dynamic models of different complexity and analysis of real data from the Amazon, North America, and Russia, we show that the reach cutoffs, besides providing stability and self-confinement to the meander belt, also act as a dynamical filter on several hydrodynamic mechanisms, selecting only those that really dominate the long-term dynamics. The results show that the long-term equilibrium conditions are essentially governed by only one spatial scale (proportional to the ratio of the river depth and the friction coefficient) and one temporal scale (proportional to the square of the spatial scale divided by the river width, the mean longitudinal velocity, and the erodibility coefficient) that contain the most important fluid dynamic quantities. The ensuing statistical long-term behavior of meandering rivers proves to be universal and largely unaffected by the details of the fluid dynamic processes that govern the short-term river behavior.