Schnorr without R in the challenge
So the scheme you suggest is that signatures are pairs (R, s) such that sG = R + H(P,m)P, where P is the public key, and m is the message.
Pick a random s. Set R = sG - H(P,m)P. (R,s) is now a valid signature for P, without knowing the corresponding private key. So this is a trivially forgeable signature scheme.
In ECDSA, the extraction of R's x coordinate, used as a scalar, plays the role of commitment in the signature to R.