Short gamma-ray burst (sGRB) jets form in the aftermath of a neutron star merger, drill through disk winds and dynamical ejecta, and extend over four to five orders of magnitude in distance before breaking out of the ejecta. Below are the first 3D general-relativistic magnetohydrodynamic sGRB simulations to span this enormous scale separation. They feature three possible outcomes: jet+cocoon, cocoon, and neither. Typical sGRB jets break out of the dynamical ejecta if (i) the bound ejecta’s isotropic equivalent mass along the pole at the time of the BH formation is <1e−4 solar mass, setting a limit on the delay time between the merger and BH formation, otherwise, the jets perish inside the ejecta and leave the jet-inflated cocoon to power a low-luminosity sGRB; (ii) the postmerger remnant disk contains strong large-scale vertical magnetic field, & 1e15 G; and (iii) if the jets are weak (<1e50 erg), the ejecta’s isotropic equivalent mass along the pole must be small (<1e−2 solar mass). Generally, the jet structure is shaped by the early interaction with disk winds rather than the dynamical ejecta. As long as our jets break out of the ejecta, they retain a significant magnetization (<1), suggesting that magnetic reconnection is a fundamental property of sGRB emission.