We show that minimal, fermionic dark matter (DM) models in the $n$-dimensional representation of the weak force with zero hypercharge that make up 100% of the DM under the standard cosmological history are strongly disfavored for $n<9$. This includes the thermal wino, which we show is ruled out even allowing for DM core sizes up to $$6.7 kpc with the preferred local DM density or $$3.7 kpc in addition to the local DM density being half the preferred value, at less than $0.2$ GeV/cm${}^3$. We reach these conclusions through dedicated searches with 14 years of Fermi gamma-ray data in the inner Galaxy between 30 GeV and 2 TeV for the continuum gamma-rays produced in the decays of unstable particles produced in DM annihilation and bound-state formation processes. We consider a variety of Milky Way DM profiles in our analyses, including those motivated by modern hydrodynamic cosmological simulations, and show that all the $n<9$ minimal DM models are disfavored even under the most conservative assumptions for these density profiles. While wino, quintuplet ($n=5$), and $n=7$ DM models are strongly disfavored by our analyses under the standard cosmology, we discuss how non-standard cosmological histories or DM sub-fractions could still allow for these particles to be realized in nature, with discovery opportunities at next-generation particle colliders and gamma-ray telescopes.