  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. We
present 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 $ \lesssim10^{-4}~{\rm M_{\odot}} $, 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 post-merger remnant disk contains strong
large-scale vertical magnetic field, $\gtrsim10^{15}$ G; and (iii) if the jets
are weak ($\lesssim10^{50}$ erg), the ejecta's isotropic equivalent mass along
the pole must be small ($\lesssim10^{-2}~{\rm M_{\odot}}$). 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 ($\lesssim1$), suggesting that magnetic reconnection
is a fundamental property of sGRB emission. The angular structure of the
outflow isotropic equivalent energy after breakout consistently features a flat
core followed by a steep power-law distribution (slope $\gtrsim3$), similar to
hydrodynamic jets. In the cocoon-only outcome, the dynamical ejecta broadens
the outflow angular distribution and flattens it (slope $\sim1.5$).
