In ECDSA, why are u and v dividing by s?
In the actual ECDSA, $(r,s)$ is called the signature of the message $m$ whose hash is $z$ i.e. $z=H(m)$.
By removing $z$ from the equation, you have completely eliminated any correlation of this "new incorrect signature" (which isn't even a signature) with the message, which is completely against the notion of a digital signature being a representative of the message which has been signed. If digital signatures did not depend on the messages, then anyone could just strip off a signature by someone, say $A$, from one message and use this signature to sign any document of their choice without $A$ even knowing.
In ECDSA, it is the $s$ value which is coupled with the message hash $z$, so if the $s$ value is completely independent of $z$ (and therefore $m$), then (r,s) is not at all a signature. It is then just a random linear equation between 2 points on an elliptic curve.