Proof that total computable functions are not enumerable
$g$ is total computable by definition.


By assumption $f : \mathbb{N}^2 \to \mathbb{N}$ is total computable.
$1$ certainly is.
$+$ certainly is.
The concatenation of $+$ and $f$ is computable by construction, and that's $g$.


The details depend on the exact definition of "computable" (i.e. the machine model) you use; see e.g. here.

So under the assumption that such $f$ existed, $g$ is computable; therefore, it must have an index in $f$, which is impossible (as you note); therefore, the assumption must be false.