Coherent vortices in two-dimensional turbulence induce far-field effects that stabilize vorticity filaments and inhibit the generation of new vortices. We show that the large-scale energy sink often included in numerical simulations of statistically stationary two-dimensional turbulence reduces the stabilizing role of the vortices, leading to filament instability and to continuous formation of new coherent vortices. This counterintuitive effect sheds new light on the mechanisms responsible for vortex formation in forced-dissipated two-dimensional turbulence, and it has significant impact on the temporal evolution of the vortex population in freely decaying turbulence. The time dependence of vortex statistics in the presence of a large-scale energy sink can be approximately described by a modified version of the scaling theory developed for small-scale dissipation.