In the generic sense of an abstract group, this is a problem since addition may not be defined. However, when working modulo a prime $p$, addition is certainly defined. However, it is not secure. In order to see why, note that we must work in a prime subgroup of $p$ in order for ElGamal to be secure. Thus, we typically choose $p=2q+1$ where $q$ is also a prime, and then we work in the subgroup of order $q$ of quadratic residues. When you add, you may get out of the subgroup.
A description of an attack when addition is used A Simple Attack on ElGamal Public Key Encryption by Dan Boneh (the paper deals with something else, but also considers addition as motivation).