  The recent report of a half-quantized thermal Hall effect in the Kitaev
material $\alpha$-RuCl$_3$ has sparked a strong debate on whether it is
generated by Majorana fermion edge currents or whether other more conventional
mechanisms involving magnons or phonons are at its origin. A more direct
evidence for Majorana fermions which could be expected to arise from a
contribution to the longitudinal heat conductivity $\kappa_{xx}$ at
$T\rightarrow0$ is elusive due to a very complex magnetic field dependence of
$\kappa_{xx}$. Here, we report very low temperature (below 1~K) thermal
conductivity ($\kappa$) of another candidate Kitaev material,
Na$_2$Co$_2$TeO$_6$. The application of a magnetic field along different
principal axes of the crystal reveals a strong directional-dependent
magnetic-field ($\bf B$) impact on $\kappa$. We show that no evidence for
mobile quasiparticles except phonons can be concluded at any field from 0~T to
the field polarized state. In particular, severely scattered phonon transport
is observed across the $B-T$ phase diagram, which is attributed to prominent
magnetic fluctuations. Cascades of phase transitions are uncovered for all $\bf
B$ directions by probing the strength of magnetic fluctuations via a precise
record of $\kappa$($B$). Our results thus rule out recent proposals for
itinerant magnetic excitations in Na$_2$Co$_2$TeO$_6$, and emphasise the
importance of discriminating true spin liquid transport properties from
scattered phonons in candidate materials.
