(Bi$_{1-x}$Sb$_x$)$_2$Te$_3$ topological insulators (TIs) are gathering increasing attention owing to their large charge-to-spin conversion efficiency and the ensuing spin-orbit torques (SOTs) that can be used to manipulate the magnetization of a ferromagnet (FM). The origin of the torques, however, remains elusive, while the implications of hybridized states and the strong material intermixing at the TI/FM interface are essentially unexplored. By combining interface chemical analysis and spin-transfer ferromagnetic resonance (ST-FMR) measurements, we demonstrate that intermixing plays a critical role in the generation of SOTs. By inserting a suitable normal metal spacer, material intermixing is reduced and the TI properties at the interface are largely improved, resulting in strong variations in the nature of the SOTs. A dramatic enhancement of a field-like torque, opposing and surpassing the Oersted-field torque, is observed, which can be attributed to the non-equilibrium spin density in Rashba-split surface bands and to the suppression of spin memory loss.