# Slight modification of El Gamal signature scheme is not secure?

A slight modification of El Gamal signature scheme would be to compute:

$$r = g^k$$ for some random k (as usual) but $$s$$ as:

$$s = (hash(m) - x)k^{-1} \mod (p-1)$$ (instead of $$(hash(m)-xr)k^{-1} \mod p - 1$$).

The verification procedure would be to check:

$$g^{hash(m)} = g^x r^s \mod p$$

I don't see what we are losing by doing so.

Forging an arbitrary ElGamal signature means finding $$(r,s)$$ such that $$g^h = y^r r^s \mod p$$ for given values of $$p$$, $$g$$, $$h$$ and $$y$$. We don't know how to do that other than selecting $$r$$ and then having to solve a discrete logarithm for $$s$$.
In your proposed variant, forging a signature means finding $$(r,s)$$ such that $$g^h = y r^s$$. This is just a matter of taking $$s = 1$$ and $$r = g^h / y$$. This can't be fixed by adding another constraint on $$s$$ since the problem would then be to find an $$n$$th root, which is solvable.
The gist of ElGamal variants is to involve a linear combination of $$r$$ and $$g^r$$ during verification. This is what makes a forgery likely to be as hard as solving a discrete logarithm.