I think I've worked out an approach to this which I find enlightening, though technically a bit fiddly.

**Claim 1:** Let $A$ be an $E_\infty$ space and let $B = K(G,n)$ be an Eilenberg-MacLane space where $G$ is abelian. Then any map of $H$-spaces $f: A \to B$ lifts to an $E_\infty$ map $f: A \to B$.

**Corollary 2:** Let $A$ be a spectrum and let $B$ be an Eilenberg-MacLane spectrum. Then any additive cohomology operation $A^\ast \Rightarrow B^\ast$ is stable.

**Proof:** By shifting and taking connective covers if necessary, we may represent $A$ by an $E_\infty$-space and $B$ by an Eilenberg-MacLane space. Then the additivity of the operation means that it is represented by a map of $H$-spaces. By Claim 1, this lifts to a map of $E_\infty$-spaces, i.e. a map of spectra, and so is stable.

**(Key) Construction 3:** If $B = K(G,n)$ is an Eilenberg-MacLane space, we may model $B$ via an $n+1$-coskeletal simplicial set which has a unique $m$-cell for each $m < n$, a unique $n$-cell for each $g \in G$, a unique $n+1$-cell for each relation in $G$. This simplicial set is a Kan complex. Thus our map $f: A \to B$ may be represented by a map of simplicial sets into this particular model of $B$.

**Lemma 4:** Model $B$ as in Construction 3. Then any pointed maps into $B$ related by a pointed homotopy are equal.

**Proof:** Let $\phi,\psi: X \to B$ be maps related by a pointed homotopy $H$. Necessarily $\phi$ and $\psi$ coincide on the $n-1$-skeleton of $X$. By the pointedness of $H$, its components at any cell of dimension $\leq n-1$ are trivial. Therefore the component of $H$ at any cell $x \in X_n$ exhibits $\phi(x),\psi(x)$ as homotopy rel their boundaries. By construction of $B$, this implies that $\phi(x) = \psi(x)$. Then $B$ is suitably coskeletal so that we must in fact have $\phi = \psi$.

**Proof of Claim 1:** Let $f: A \to B$ be a map of $H$-spaces, represented where $B$ is the simplicial set from Construction 3. Then there is a pointed homotopy between $\mu \circ f^{\times n}$ and $f \circ \mu$ (where $\mu$ ambiguously denotes the multiplication on $A$ or $B$). By Lemma 4, we have $\mu \circ f^{\times n} = f \mu$. Thus the constant homotopy is a $\Sigma_n$-equivariant homotopy between these two maps. Passing to $\Sigma_n$-homotopy fixed points, we obtain diagram as follows, where the bottom square and the outer rectangle commute up to pointed homotopy:

$$ \require{AMScd} \begin{CD} E\Sigma_n \times_{\Sigma_n} A^n @>E\Sigma_n \times_{\Sigma_n} f^n>> E\Sigma_n \times_{\Sigma_n} B^n\\ @VVV  @VVV\\ A @>f>> B \\ @VVV  @VVV\\ B\Sigma_n \times A @>id \times f>> B\Sigma_n \times B \end{CD} $$

The map $B \to B\Sigma_n \times B$ is a split monomorphism, so the top square commutes up to pointed homotopy as well. Invoking Lemma 4 again, the top square commutes strictly. Since this is true for every $n$, we have that $f$ is a map of $E_\infty$-spaces as desired.