  Indirect detection opens a unique window for probing thermal dark matter
(DM): the same annihilation process that determined the relic abundance in the
early Universe drives the present day astrophysical signal. While TeV-scale
particles weakly coupled to the Standard Model face undoubted challenges from
decades of null searches, the scenario remains compelling, and simple
realizations such as Higgsino DM remain largely unexplored. The fate of such
scenarios could be determined by gamma-ray observations of the centre of the
Milky Way with Imaging Atmospheric Cherenkov Telescopes (IACTs). We consider
the ultimate sensitivity of current IACTs to a broad range of TeV-scale DM
candidates - including specific ones such as the Wino, Higgsino, and
Quintuplet. To do so, we use realistic mock H.E.S.S.-like observations of the
inner Milky Way halo, and provide a careful assessment of the impact of recent
Milky Way mass modeling, instrumental and astrophysical background
uncertainties in the Galactic Center region, and the theoretical uncertainty on
the predicted signal. We find that the dominant systematic for IACT searches in
the inner Galaxy is the unknown distribution of DM in that region, however,
beyond this the searches are currently statistically dominated indicating a
continued benefit from more observations. For two-body final states at $1~{\rm
TeV}$, we find a H.E.S.S.-like observatory is sensitive to $\langle \sigma v
\rangle \sim 3 \times 10^{-26}-4 \times 10^{-25}~{\rm cm}^3{\rm s}^{-1}$,
except for neutrino final states, although we find results competitive with
ANTARES. In addition, the thermal masses for the Wino and Quintuplet can be
probed; the Higgsino continues to be out of reach by at least a factor of a
few. Our conclusions are also directly relevant to the next generation
Cherenkov Telescope Array, which remains well positioned to be the discovery
instrument for thermal DM.
