  The superconducting state of the heavy-fermion metal UTe$_2$ has attracted
considerable interest because of evidence for spin-triplet Cooper pairing and
non-trivial topology. Progress on these questions requires identifying the
presence or absence of nodes in the superconducting gap function and their
dimension. In this article we report a comprehensive study of the influence of
disorder on the thermal transport in the superconducting state of UTe$_2$.
Through detailed measurements of the magnetic field dependence of the thermal
conductivity in the zero-temperature limit, we obtain clear evidence for the
presence of point nodes in the superconducting gap for all samples with
transition temperatures ranging from 1.6~K to 2.1~K obtained by different
synthesis methods, including a refined self-flux method. This robustness
implies the presence of symmetry-imposed nodes throughout the range studied,
further confirmed via disorder-dependent calculations of the thermal transport
in a model with a single pair of nodes. In addition to capturing the
temperature dependence of the thermal conductivity up to $T_c$, this model
allows us to limit the possible locations of the nodes, suggesting a B$_{1u}$
or B$_{2u}$ symmetry for the superconducting order parameter. Additionally,
comparing the new, ultra-high conductivity samples to older samples reveals a
crossover between a low-field and a high field regime at a single value of the
magnetic field in all samples. In the high field regime, the thermal
conductivity at different disorder levels differ from each other by a simple
offset, suggesting that some simple principle determines the physics of the
mixed state, a fact which may illuminate trends observed in other clean nodal
superconductors.
