There are already several excellent questions and answers on MO regarding Steenrod squares, understanding them in various ways and relating them to power operations and I think I get this. Still, I am confused about the sense in which the power operation perspective coincides with the stable cohomology operation perspective. Namely, there seems to be two very different constructions that for the Eilenberg-MacLane spectrum $H\mathbb{F}_2$ yield (almost) the same answer.

1) Given a spectrum $E$, we can consider the graded (non-commutative) ring $\mathcal{A}_E =[E,E]_*$ with multiplication induced by composition. For every spectrum $X$, the ring $\mathcal{A}_E$ acts on the graded abelian group $E^*(X)=[X,E]_*$. For $E=H\mathbb{F}_2$, we get the *ordinary Steenrod algebra* which is generated by the $Sq^i$ operations with $i\ge 0$. This can be calculated explicitly as

$$\mathcal{A}_2^*=\lim_nH^{*+n}(K(\mathbb{F}_2,n);\mathbb{F}_2)$$

and the Borel-Serre computation of the cohomology of Eilenberg-MacLane spaces (I mention this to emphasis that, as far as I understand, the presentation with $Sq^i$-s is obtained by *computation*)

2) Given an $E_\infty$-ring spectrum $R$ we can consider the homotopy groups of the endomorphism spectrum of the forgetful functor from $E_\infty$-algebras over $R$ to spectra. Less formally, the thing that naturally acts on the underlying spectrum of every $R$-algebra. This includes the power operations for example. For $R = H\mathbb{F}_2$, we get the *big Steenrod algebra* (more precisely, a certain completion of it. See Lurie's notes), which is generated (topologically) by all Steenrod squares $Sq^i$ with $i\in\mathbb{Z}$. The ordinary Steenrod algebra can be obtained from it by imposing the single relation $Sq^0 = Id$.

In general, the two constructions take different things as input and produce an algebra that in general acts on different things. Yet for $H\mathbb{F}_2$, we get closely related algebras.

Question: Is there a conceptual explanation of the relation between (1) and (2)? Is there anything more general we can say about this relation for (ring) spectra other then $H\mathbb{F}_p$?

Following Lurie's notes cited above, it seems that the main step in the comparison is the fact that the $\mathbb{F}_2$-cochains of $K(\mathbb{F}_2,n)$ as an algebra over $H\mathbb{F}_2$ is *almost* the free algebra on one generator of degree $n$, where the only relation we need to impose is precisely $Sq^0=Id$. But again, this is verified in a very computational way and from this perspective seems to me like a miracle. Another point is that power operations for other cohomology theories (like K-theory) are unstable in general, so this also seems rather special.