  The observation of time-reversal symmetry breaking and large residual density
of states in tetragonal FeSe$_{1-x}$S$_x$ suggests a novel type of ultranodal
superconducting state with Bogoliubov Fermi surfaces (BFSs). Although such BFSs
in centrosymmetric superconductors are expected to be topologically protected,
the impurity effect of this exotic superconducting state remains elusive
experimentally. Here, we investigate the impact of controlled defects
introduced by electron irradiation on the superconducting state of tetragonal
FeSe$_{1-x}$S$_x$ ($0.18\le x\le 0.25$). The temperature dependence of magnetic
penetration depth is initially consistent with a model with BFSs in the
pristine sample. After irradiation, we observe a nonmonotonic evolution of
low-energy excitations with impurity concentrations. This nonmonotonic change
indicates a transition from nodal to nodeless, culminating in gapless with
Andreev bound states, reminiscent of the nodal $s_\pm$ case. This points to the
accidental nature of the possible BFSs in tetragonal FeSe$_{1-x}$S$_x$, which
are susceptible to disruption by the disorder.
