Is the uniqueness of the additive neutral element sufficient to prove x+z=x implies z=0?

No, this cannot be proved from just associativity, commutativity, and existence of a neutral element. For instance, consider the set $[0,1]$ with the binary operation $a*b=\min(a,b)$. This operation is associative and commutative and $1$ is a neutral element. But for any $x,y$ with $x\leq y$, we have $x*y=x$, and $y$ is not necessarily the neutral element $1$.

For an example with a more additive flavor, let's extend the operation $+$ to a new element $\infty$ with the rule that $x+\infty=\infty+x=\infty$ for all $x$. You can check that $+$ is still associative and commutative, and $0$ is still its identity element. However, we have $\infty+7=\infty$ and $7\neq0$.