Mapping the hash of message to a point of elliptic curve for signature
That's insecure.
In BLS signatures: for private key $x$ and public key $X = xP$, the signature is computed as $T = xS$, and the verification checks if $e(T, P) = e(S, X)$, which works because:

$e(T, P) = e(xS, P) = e(xS, P) = e(S, P)^x$
$e(S,X) = e(S, xP) = e(S, P)^x$

If you know that $S = kP$, then you can forge a signature for a message with hash $k'$ from a signature of a message with hash $k$ by simply computing $T' = k'k^{-1}T$, which works because:

$T' = k'k^{-1}xS = k'k^{-1}xkP = xk'P = xS'$