  A recent study [S. Yoshizawa {\it et al}., Nature Communications {\bf 12},
1462 (2021)] reported the occurrence of field-resilient superconductivity, that
is, enhancement of the in-plane critical magnetic field $H^{||}_{\rm c2}$
beyond the paramagnetic limiting field, in atomic-layer crystalline
($\sqrt{7}\times\sqrt{3}$)-In on a Si(111) substrate. The present article
elucidates the origin of the observed field-resilient noncentrosymmetric
superconductivity in this highly crystalline two-dimensional material. We
develop the quasiclassical theory of superconductivity by incorporating the
Fermi surface anisotropy together with an anisotropic spin splitting and
texture specific to atomic-layer crystalline systems. In
Si(111)-($\sqrt{7}\times\sqrt{3}$)-In, a typical material with a large
antisymmetric spin-orbit coupling (ASOC), we show an example where the
combination of the ASOC and disorder effect suppresses the paramagnetic
depairing and can lead to an enhancement of $H^{||}_{\rm c2}$ compared to an
isotropic system only when a magnetic field is applied in a particular
direction due to an anisotropic spin texture. We also study the parity-mixing
effect to demonstrate that the enhancement of $H^{||}_{\rm c2}$ is limited in
the moderately clean regime because of the fragile $s$+$p$-wave pairing against
nonmagnetic scattering in the case of the dominant odd-parity component of a
pair wavefunction. Furthermore, from analysis of the transition line, we
identify the field-resilience factor taking account of the scattering and
suppression of paramagnetic effects and discuss the origin of the
field-resilient superconductivity. Through fitting of the $H^{||}_{\rm c2}$
data, the normal-state electron scattering is discussed with a prime focus on
the role of atomic steps on a Si(111) surface.
