Pressure-driven flows of aqueous ionic liquids are characterized by electroviscosity an increase in the effective (apparent) viscosity because of an induced back electric field termed streaming potential. In this work, we investigate the electrokinetic phenomenon of streaming potential mediated flows in ion-selective nanopores. We report a dramatic augmentation in the effective viscosity as attributable to the finite size effect of the ionic species in counterion-only systems. The underlying physics involves complex interaction between the concerned electrochemical phenomena and hydrodynamic transport in a confined fluidic environment, which we capture through a modified continuum based approach and validate using molecular dynamics simulations. We obtain an expression for the ionic-size dependent streaming potential pertinent to the physical situation being addressed. The corresponding estimations of effective viscosity implicate that the classical paradigm of point sized ions can give rise to gross underestimations of the flow resistance in counterion-only systems especially for negligible surface (Stern layer) conductivity and large fluidic slip at the surface.