  { {\bf Background:} Knowing the difference between the neutron and proton
densities of nuclei is a significant topic because of its importance for
understanding neutron star structures and cooling mechanisms. The
coherent-nuclear photoproduction of pions, $({\g,\pi^0})$, combined with
elastic electron scattering, has been suggested to be a very accurate probe of
density differences. {\bf Purpose:} Study the ${(\g,\pi^0})$ reaction mechanism
so as to better access the uncertainties involved in extracting the neutron
density. {\bf Methods:} Include the effects of final-state pion-nucleus
charge-exchange reactions on the cross section and study the influence of the
non-zero spatial extent of the proton. {\bf Results:} The effects of
final-state charge-exchange increase the cross section between 6\% and 5\%,
generally decreasing as the momentum transfer increases. This leads to an
increase of the extracted neutron skin distance by about 50\%. The validity of
the previous treatments of the proton size is confirmed. {\bf Conclusion:} The
model dependence of the theoretically computed cross section increases the
total systematic uncertainty (experiment plus theory) in extracting the neutron
skin from the ${(\g,\pi^0})$ cross section by at least a factor of three. }
