There is a point of view on the Steenrod algebra that goes something like the following: the functor $-\otimes H\mathbb{F}_p\colon Mod_{\mathbb{S}}\to Mod_{H\mathbb{F}_p}$ corresponds to pulling back quasicoherent sheaves on $Spec(\mathbb{S})$ to the point $Spec(H\mathbb{F}_p)$, i.e. a sort of "fiber functor" that takes the fiber of a spectrum at this point. One may then ask what the endomorphisms of this fiber functor are and of course one recovers the Steenrod algebra. Being a Hopf-algebra, the Steenrod algebra corresonds to an affine (super) group scheme and we can ask how close its "representations" are to $\mathbb{S}$-modules themselves. Trying to rebuild spectra from representations of the Steenrod algebra is essentially the Adams spectral sequence.
This is closely related to (essentially the same point of view as) thinking of the descent coring $\mathbb{D}_p=H\mathbb{F}_p\otimes H\mathbb{F}_p$ (really the entire Amitsur complex) for the map $\mathbb{S}\to H\mathbb{F}_p$, so that descent data are $H\mathbb{F}_p$-modules with $\mathbb{D}_p$-comodule structures. Then descent corresponds to taking the cofixed points (i.e. the homotopy equalizer of the coaction with the trivial coaction, using that $\mathbb{D}_p$ is coaugmented) of the descent datum, whose Bousfield-Kan spectral sequence is the Adams spectral sequence. Again we can ask whether or not the canonical functor $Mod_\mathbb{S}\to Desc(\mathbb{S}\to H\mathbb{F}_p)$ is an equivalence.
In general, of course, neither comparison is an equivalence. For one, descending in either case gives a $p$-complete spectrum. Additionally, you're totally hosed if you try to deal with nonconnective spectra, among other things. But the point, at the end of the day, is that both of these are "Tannakian reconstruction" approaches to understanding $\mathbb{S}$-modules.
This is all buildup to a question which is really about motives and motivic stable homotopy theory, about which I know next to nothing. In light of the above perspective, we might attempt to recover the stable motivic homotopy category from representations of the motivic Steenrod algebra (or comodules over the dual motivic Steenrod algebra). On the other hand, the "classical" constructions of the motivic Galois group that I've seen (about which I understand very little) seem to also take a Tannakian approach. Indeed, at least one case appears to consider endomorphisms of Betti cohomology which is (I believe) a motivic incarnation of singular cohomology. I'm certain that there are many finer points and distinctions this is all missing. I'm basically doing low-level pattern matching. But the question ultimately is:
Is any part of the classical motivic Galois group visible in the motivic Steenrod algebra?
