The report of a half-quantized thermal Hall effect and oscillatory structures in the magnetothermal conductivity in the Kitaev material α-RuCl₃ have sparked a strong debate on whether it is generated by Majorana fermion edge currents, spinon Fermi surface, or whether other more conventional mechanisms are at its origin. Here, we report low temperature thermal conductivity (κ) of another candidate Kitaev material, Na₂Co₂TeO₆. The application of a magnetic field (B) along different principal axes of the crystal reveals a strong directional-dependent B impact on κ, while no evidence for mobile quasiparticles except phonons can be concluded at any field. Instead, severely scattered phonon transport prevails across the B−T phase diagram, revealing cascades of phase transitions for all B directions. Our results thus cast doubt on recent proposals for significant itinerant magnetic excitations in Na₂Co₂TeO₆, and emphasize the importance of discriminating true spin liquid transport properties from scattered phonons in candidate materials.