Why the auxiliary field cannot be discarded off shell

After elimination, $s_0\bar{c}=iD\gamma$, and therefore

This vanishes on the free ghost shell, not on unrestricted ghost configurations. We retain $b$ in the off-shell multiplet.

For bookkeeping, name the antifields

An antifield has opposite parity and ghost number $-1-\mathrm{gh}(\text{field})$. These names denote external source slots only. They are not additional propagating generators here; no antifield differential, BV antibracket, or interacting master action is claimed.