Is there a problem with this non-ECDSA message signing? Is there a problem with signing a message like this?

p = the Elliptic Curve prime  
G = base point  
a = Alice’s private key  
A = aG = Alice’s public key  
e = hash(message)  
Jx = x coordinate of J  


In my target 32-bit microcontroller I am using p-256 and SHA256.  Thus e will typically be in 1..p-1.

Alice computes:

e = hash(message)
E = eG
J = aE          using Alice’s private key


and sends Jx as signature

Recipient computes:

e = hash(message)  
K = eA          using Alice’s public key
and validates Jx = Kx 


The only problem I see is that e might be 0 or 1 or some value >= p.  (If it is I have the flexibility to pad the message such that e is in a good range.)  

Is there a problem if hash(msg) is greater or equal to the EC prime?  Other than a few unusable values of e is there is a problem that makes this method less secure than ECDSA?
The problem with this method is that it is insecure.  Anyone with Alice's public key can sign any message they want, by computing:

$e = hash(message)$

$J = eA$

This $J_x$ will verify correctly according to your verification procedure.