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.