  An examination is conducted on a commonly used procedure for extracting
(un)polarized $\gamma n\to \pi^-p$ and $\gamma n\to \pi^0n$ observables from
$d(\gamma,\pi^-)pp$ and $d(\gamma,\pi^0)pn$ data, using a model that consists
of the impulse term and the final-state interaction (FSI) terms due to nucleon-
and pion-exchange. Recent experimental and theoretical analyses used an
extraction method that does not impose a cut on the final $\pi N$ invariant
mass $W$. I demonstrate that the use of this method can result in the $\gamma
n\to \pi N$ observables that are seriously distorted by the nucleon Fermi
motion, and that one can efficiently avoid this problem by imposing a cut on
$W$. It is also shown that the use of kinematical cuts of recent experimental
analyses can still leave in the selected samples substantial FSI effects that
must be corrected in extracting the $\gamma n \to\pi N$ cross sections. In
terms of the nucleon- and pion-exchange mechanisms, I give the first
qualitative explanation of the FSI corrections, obtained in a recent MAMI
experiment, for extracting $\gamma n\to \pi^0 n$ cross sections.
