  \noindent We revisit the \textit{minimal seesaw model}, i.e., the type-I
seesaw mechanism involving only two right-handed neutrinos. % This model
represents an important minimal benchmark scenario for future experimental
updates on neutrino oscillations. % It features four real parameters that
cannot be fixed by the current data: two $CP$-violating phases, $\delta$ and
$\sigma$, as well as one complex parameter, $z$, that is experimentally
inaccessible at low energies. % The parameter $z$ controls the structure of the
neutrino Yukawa matrix at high energies, which is why it may be regarded as a
label or index for all UV completions of the minimal seesaw model. % The fact
that $z$ encompasses only two real degrees of freedom allows us to
systematically scan the minimal seesaw model over all of its possible UV
completions. % In doing so, we address the following question: Suppose $\delta$
and $\sigma$ should be measured at particular values in the future---to what
extent is one then still able to realize approximate textures in the neutrino
Yukawa matrix? % Our analysis, thus, generalizes previous studies of the
minimal seesaw model based on the assumption of exact texture zeros. % In
particular, our study allows us to assess the theoretical uncertainty inherent
to the common texture ansatz. % One of our main results is that a normal
light-neutrino mass hierarchy is, in fact, still consistent with a two-zero
Yukawa texture, provided that the two texture zeros receive corrections at the
level of $\mathcal{O}\left(\textrm{10}\,\%\right)$. % While our numerical
results pertain to the minimal seesaw model only, our general procedure appears
to be applicable to other neutrino mass models as well.
