ElGamal signature scheme: Why is r included in s
With your proposed modification of the ElGamal signature scheme you can produce forgeries for arbitrary (hashed) messages $m$. 

By looking at the verification equation $$g^m = yr^s$$ you just have to set $r$ to $r=(g^my^{-1})^{s^{-1}}$ (just by rearranging the verification equation) which you can do for any $s$ from $\mathbb{Z}_{p-1}^*$, i.e., every $s$ that has a multiplicative inverse in $\mathbb{Z}_{p-1}$.