To start, let's work with mod $p$ cohomology $H\mathbb F_p$ where $p$ is a prime. Consider the following three things:

1. The bigraded abelian group of all unstable cohomology operations, comprising all natural transformations of set-valued functors $H^m(-;\mathbb F_p) \Rightarrow H^n(-;\mathbb F_p)$, i.e. $$Unst(H\mathbb F_p) = (\pi_0 Map(K(\mathbb F_p,m), K(\mathbb F_p,n)))_{m,n \in \mathbb N}$$

2. The bigraded abelian group of additive cohomology operations, i.e. the subgroup $Unst^{add}(H\mathbb F_p) \subset Unst(H\mathbb F_p)$ comprising natural transformations of abelian group-valued functors $H^m(-;\mathbb F_p) \Rightarrow H^n(-;\mathbb F_p)$.

3. The bigraded abelian group of all stable cohomology operations $St(H\mathbb F_p) = (\pi_0 Map (\Sigma^m H \mathbb F_p, \Sigma^n H\mathbb F_p))_{m,n \in \mathbb N} \subseteq Unst^{add}$. Of course we have $St(H\mathbb F_p)_{m,n} = \mathcal A^{n-m}$ where $\mathcal A^\ast = \pi_{-\ast} Map(H\mathbb F_p,H \mathbb F_p)$ is the Steenrod algebra.

Now, the standard calculation of $\mathcal A^\ast$ proceeds by calculating $Unst(H\mathbb F_p)$. Looking at the results, I believe the answer to the following question is affirmative:

Question 1: Do we have $Unst^{add}(H\mathbb F_p) = St(H\mathbb F_p)$?

For instance, when $n$ is not a power of $p$ the $n$th-power operation $(-)^n: H^m(-;\mathbb F_p) \Rightarrow H^{nm}(-;\mathbb F_p)$ is not a stable operation, but this is already explained by the fact that it is not an additive operation.

Assuming I have it right and the answer to Question 1 is "yes", I have some follow-up questions:

Question 2: Is there a "good reason" why $Unst^{add}(H\mathbb F_p) = St(H\mathbb F_p)$?

Question 3: Is this a general fact? Is it the case that for any spectrum / generalized cohomology theory $E$ we have $Unst^{add}(E) = St(E)$?

I expect the answer to Question 3 is "no" in this generality, but we can ask:

Question 4: Do we have $Unst^{add}(E) = St(E)$ whenever $E$ is a sum of Eilenberg-MacLane spectra?

As a small bit of evidence that this might be a general phenomenon, note that if a cohomology operation $\phi: K(A,m) \to K(B,n)$ is additive, then by the Yoneda lemma it corresponds to a map of abelian group objects in the homotopy category. This looks like at least the first step to showing that $\phi$ is a map of infinite loop spaces, and thus lifts to a map of spectra. It would be really convenient if every map between topological abelian groups which is a map of abelian group objects in the homotopy category could be rectified to a map of topological abelian groups, but this is false: it would imply that every additive cohomology operation between sums of Eilenberg-MacLane spectra would be a map of $H\mathbb Z$-modules. Counterexamples are given e.g. by every Steenrod power operation except for $Sq^1$ which coincides with the Bockstein.

To start, let's work with mod $p$ cohomology $H\mathbb F_p$ where $p$ is a prime. Consider the following three things:

1. The bigraded abelian group of all unstable cohomology operations, comprising all natural transformations of set-valued functors $H^m(-;\mathbb F_p) \Rightarrow H^n(-;\mathbb F_p)$, i.e. $$Unst(H\mathbb F_p) = (\pi_0 Map(K(\mathbb F_p,m), K(\mathbb F_p,n)))_{m,n \in \mathbb N}$$

2. The bigraded abelian group of additive cohomology operations, i.e. the subgroup $Unst^{add}(H\mathbb F_p) \subset Unst(H\mathbb F_p)$ comprising natural transformations of abelian group-valued functors $H^m(-;\mathbb F_p) \Rightarrow H^n(-;\mathbb F_p)$.

3. The bigraded abelian group of all stable cohomology operations $St(H\mathbb F_p) = (\pi_0 Map (\Sigma^m H \mathbb F_p, \Sigma^n H\mathbb F_p))_{m,n \in \mathbb N}$. There is a natural map $St(H\mathbb F_p) \to Unst^{add}$. Of course we have $St(H\mathbb F_p)_{m,n} = \mathcal A^{n-m}$ where $\mathcal A^\ast = \pi_{-\ast} Map(H\mathbb F_p,H \mathbb F_p)$ is the Steenrod algebra. So there is also a natural map $St(H\mathbb F_p)_{0,k} \to \prod_{n-m = k} Unst^{add}(H\mathbb F_p)_{m,n}$.

Now, the standard calculation of $\mathcal A^\ast$ proceeds by calculating $Unst(H\mathbb F_p)$. Looking at the results, I believe the answer to the following questions are affirmative:

Question 1: Is the natural map $St(H\mathbb F_p) \to Unst^{add}(H\mathbb F_p)$ a surjection?

Question 1': Is the natural map $St(H\mathbb F_p)_k \to \prod_{n-m = k} Unst^{add}(H\mathbb F_p)_{m,n}$ an injection?

For instance, when $n$ is not a power of $p$ the $n$th-power operation $(-)^n: H^m(-;\mathbb F_p) \Rightarrow H^{nm}(-;\mathbb F_p)$ is not a stable operation, but this is already explained by the fact that it is not an additive operation.

Assuming I have it right and the answers to Question 1 and 1' are "yes", I have some follow-up questions:

Question 2: Is there a "good reason" for the affirmative answers to Questions 1,1'?

Question 3: Do these facts generalize to an arbitrary spectrum $E$ in place of $H\mathbb F_p$?

I expect the answer to Question 3 is "no" in this generality, but we can ask:

Question 4: Do these facts generalize to an arbitrary sum of Eilenberg-MacLane spectra $E$ in place of $H\mathbb F_p$?

As a small bit of evidence that this might be a general phenomenon, note that if a cohomology operation $\phi: K(A,m) \to K(B,n)$ is additive, then by the Yoneda lemma it corresponds to a map of abelian group objects in the homotopy category. This looks like at least the first step to showing that $\phi$ is a map of infinite loop spaces, and thus lifts to a map of spectra. It would be really convenient if every map between topological abelian groups which is a map of abelian group objects in the homotopy category could be rectified to a map of topological abelian groups, but this is false: it would imply that every additive cohomology operation between sums of Eilenberg-MacLane spectra would be a map of $H\mathbb Z$-modules. Counterexamples are given e.g. by every Steenrod power operation except for $Sq^1$ which coincides with the Bockstein.