Is the one-time pad still perfectly secret if all-zero keys are excluded?
Yes your solution is correct; the flawed OTP scheme is no longer perfectly secret if there are fewer keys than messages.

Once you remove the all zero key the property
$$
\mathbb{P}[M=m|C=c]=\mathbb{P}[M=m],\qquad \forall m,\forall c,\quad(1)
$$
required for perfect secrecy will no longer necessarily