I'm trying to understand diagonalization as it applies to proving that not all total functions are primitive recursive functions. For example, say that we enumerate all primitive recursive functions as $f_0, f_1, f_2, \dots$. Then, suppose that we show that $d(x) = f_x(x) + 1$ is not in this list (by diagonalization), and so it is not primitive recursive.
Does it then make sense to argue that $g(x) = f_x(x)$ is not primitive recursive either, because if it were, then $d(x) = g(x)+1$ would be primitive recursive as well (since the set of primitive recursive functions is closed under the successor function)?
Yes. It makes sense to say that $d$ is not primitive recursive, for precisely the reason that you state.