Why does DSA require that a new $k$ be chosen and the algorithm restarted if $s=0$?
The algorithm needs to be restarted because the private key $x$ can be obtained when $s=0$.
From:

$s=[k^{-1}( H(M) + xr)] \mod q$

We can solve for $x$ and get:

$x=\frac{-H(M)}{r} \mod q$