  We present the first numerical simulations that track the evolution of a
black hole-neutron star (BH-NS) merger from pre-merger to
$r\gtrsim10^{11}\,{\rm cm}$. The disk that forms after a merger of mass ratio
$q=2$ ejects massive disk winds ($3-5\times10^{-2}\,M_{\odot}$). We introduce
various post-merger magnetic configurations and find that initial poloidal
fields lead to jet launching shortly after the merger. The jet maintains a
constant power due to the constancy of the large-scale BH magnetic flux until
the disk becomes magnetically arrested (MAD), where the jet power falls off as
$L_j\sim t^{-2}$. All jets inevitably exhibit either excessive luminosity due
to rapid MAD activation when the accretion rate is high or excessive duration
due to delayed MAD activation compared to typical short gamma-ray bursts
(sGRBs). This provides a natural explanation for long sGRBs such as GRB 211211A
but also raises a fundamental challenge to our understanding of jet formation
in binary mergers. One possible implication is the necessity of higher binary
mass ratios or moderate BH spins to launch typical sGRB jets. For post-merger
disks with a toroidal magnetic field, dynamo processes delay jet launching such
that the jets break out of the disk winds after several seconds. We show for
the first time that sGRB jets with initial magnetization $\sigma_0>100$ retain
significant magnetization ($\sigma\gg1$) at $r>10^{10}\,{\rm cm}$, emphasizing
the importance of magnetic processes in the prompt emission. The jet-wind
interaction leads to a power-law angular energy distribution by inflating an
energetic cocoon whose emission is studied in a companion paper.
