Why we need ECDSA when we have ECDH?
To simplify your definition:

$$
s = \operatorname{sign}(k, m) = h(G\ h(m)\ k)\\
\operatorname{verify}(Gk, m, s)\iff s \overset{?}{=} h(Gk\ h(m))
$$

This is obviously insecure because I can forge "signatures"!

$$s' = \operatorname{forge}(Gk, m') = h(Gk\ h(m'))$$

ECDH is not a signature scheme. Schnorr is amazing and is much more interesting than ECDSA! Including but not limited to: threshold k-of-n distributed identities, n-of-n multisig aggregation, batch verification, public key recovery, implicit certificates, and much more!