  The detection of the gravitational wave counterpart GRB 170817A,
underluminous compared to the cosmological GRB population by a factor of
10,000, motivates significant effort in detecting and localizing a dim, nearby,
and slightly off-axis population of short GRBs. Swift/BAT is the most sensitive
GRB detector in operation, and the only one that regularly localizes GRBs to
arcminute precision, critical to rapid followup studies. However, the utility
of BAT in targeted sub-threshold searches had been historically curtailed by
the unavailability of the necessary raw data for analysis. The new availability
of time-tagged event (TTE) data from the GUANO system (arXiv:2005.01751),
motivates renewed focus on developing sensitive targeted search analysis
techniques to maximally exploit these data. While computationally cheap, we
show that the typical coded-mask deconvolution imaging is limited in its
sensitivity due to several factors. We formalize a maximum likelihood framework
for the analysis of BAT data wherein signals are forward modelled through the
full instrument response, and -- coupled with the development of new response
models -- demonstrate its superior sensitivity to typical imaging via archival
comparisons, injection campaigns, and, after implementing as a targeted search,
a large number of low-latency GRB discoveries and confirmed arcminute
localizations to date. We also demonstrate independent localization of some
out-of-FOV GRBs for the first time. NITRATES's increased sensitivity boosts the
discovery rate of GRB 170817A-like events in BAT by a factor of at least
$3-4$x, along with enabling joint analyses and searches with other GRB, GW,
neutrino, and FRB instruments. We provide public access to the response
functions and search pipeline code.
