The most popular model for short gamma-ray bursts (sGRBs) involves the coalescence of binary neutron stars. Because the progenitor is actually hidden from view, we must consider under which circumstances such merging systems are capable of producing a successful sGRB. Soon after coalescence, winds are launched from the merger remnant. In this paper, we use realistic wind profiles derived from global merger simulations in order to investigate the interaction of sGRB jets with these winds using numerical simulations. We analyze the conditions for which these axisymmetric winds permit relativistic jets to break out and produce an sGRB. We find that jets with luminosities comparable to those observed in sGRBs are only successful when their half-opening angles are below ≈20°. This jet collimation mechanism leads to a simple physical interpretation of the luminosities and opening angles inferred for sGRBs. If wide, low-luminosity jets are observed, they might be indicative of a different progenitor avenue such as the merger of a neutron star with a black hole. We also use the observed durations of sGRB to place constraints on the lifetime of the wind phase, which is determined by the time it takes the jet to break out. In all cases we find that the derived limits argue against completely stable remnants for binary neutron star mergers that produce sGRBs.