Spin Hall nano-oscillators (SHNOs) are nanoscale microwave sources with potential applications in local spin excitation, neuromorphic computing, and oscillator-based Ising machines. Further improvements in output power and coherence are essential for these applications, and increasing the thickness of the ferromagnetic layer offers a promising route toward this goal. However, this comes at the cost of higher drive currents, which increase Joule heating and can degrade device performance. Here, we show that this bottleneck can be overcome by SiC substrates. By comparing otherwise identical W/NiFe nanoconstriction SHNOs on Al2O3, Si, and SiC, we find that the high thermal conductivity of SiC enables stable auto-oscillations in NiFe layers as thick as 10 nm, while the optimum performance occurs at 7 nm. The best SiC device reaches an average output power of 13.1 pW and a mean quality factor of Q = 23,000, representing the highest reported simultaneous output power and coherence for a single free-running SHNO. These results establish SiC and thermal substrate engineering as a route to thick-ferromagnet spintronic oscillators for coherent microwave generation in nanoscale devices.