  Model uncertainties, arising due to suppression of target excitations in the
description of deuteron scattering and resulting in a modification of the
two-body interactions in a three-body system, are investigated for several
$(d,p)$ reactions serving as indirect tools for studying the astrophysical
$(p,\gamma)$ reactions relevant to $rp$-process. The three-body nature of
deuteron-target potential is treated within adiabatic distorted wave
approximation (ADWA) which relies on dominant contribution from the components
of the three-body deuteron-target wave function with small $n$-$p$ separations.
This results in a simple prescription for treating the explicit
energy-dependence of two-body optical potentials in a three-body system
requiring nucleon optical potentials to be evaluated at a shifted energy with
respect to the standard value of half the deuteron incident energy. In
addition, the ADWA allows for leading-order multiple scattering effects to be
estimated, which leads to a simple renormalization of the adiabatic potential's
imaginary part by the factor of two. These effects are assessed using both
nonlocal and local optical potential systematics for $^{26}$Al, $^{30}$P,
$^{34}$Cl and $^{56}$Ni targets at deuteron incident energy of 12 MeV typical
for experiments with radioactive beams in inverse kinematics. The model
uncertainties induced by the three-body nature of deuteron-target scattering
are found to be within 40$\%$ both in the main peak of angular distributions
and in total $(d,p)$ cross sections. At higher deuteron energies, around 60
MeV, model uncertainties can reach 100\% in the total cross sections. A few
examples of application to astrophysically interesting proton resonances in
$^{27}$Si and $^{57}$Cu obtained using $(d,p)$ reactions and mirror symmetry
are given.
