Fluctuations of the sediment volume stored in mountain channels are driven by stochastic variations of discharge and sediment supply and can inhibit bedrock incision if sediment thickness is too large. Here, I study how this short-term stochasticity propagates into the long-term reduction of bedrock incision efficiency (the cover effect) at geological time scales. I introduce a new numerical model that resolves sediment transport and bedrock incision at daily time scales, and is run for thousands of years. It incorporates (1) a transport threshold and daily stochastic variations in water discharge and sediment supply, (2) a freely evolving channel width and slope, and (3) an explicit treatment of alluvial thickness variations and corresponding bed incision reduction. For typical mountain river conditions the model predicts that alluvial cover oscillates between complete and negligible incision reduction. In this intermittent regime the long-term cover effect is mainly set by the fraction of time spent in full cover, and the present-day extent of alluvial cover is not representative of long-term dynamics. The long-term integrated cover effect law differs strongly from proposed theoretical and experimental models, and it is controlled by sediment supply stochasticity rather than the details of cover development at the hydraulic time scale. Model results also suggest that steady state channel configuration always depends on sediment supply rate, while being never limited by transport capacity or strictly detachment limited. These results point out that discharge and sediment supply stochasticity should not be considered less important than the intricate details of incision laws to model long-term bedrock channel dynamics.