Eric wrote a really nice response telling that your initial hope is incorrect and why. I'd just like to write some positive results that you can find.
Disclaimer: I understand little to nothing about the case of a compact Lie group.
Schwede and Shipley have a paper entitled "Stable model categories are categories of modules" from 2003. In particular, G-spectra form a stable model category to which their results apply. Schwede-Shipley show that if you pick a set of "generators", then you'll get an category $I$ enriched in spectra, with $Hom_I(i,j)$ being a spectrum together with units $\mathbb S \to Hom_I(i,i)$ and composition maps $$Hom_I(j,k) \wedge Hom_I(i,j) \to Hom_I(i,k)$$ which are unital and associative. (This is the spectrum version of a DG-category, if you like). Then there is an equivalence between $G$-spectra and enriched functors from $I$ to the category of spectra.
In $G$-spectra, we can pick generators given by the spectra $\Sigma^\infty G/H_+$, which are representing objects for the standard "fixed point" functors. So a $G$-spectrum is equivalent to the data of a collection of spectra $Y^H$ as $H$ ranges over the subgroups of $G$, together with "action maps"
$$F(\Sigma^\infty G/H_+, \Sigma^\infty G/K_+) \wedge Y^H \to Y^K$$
that are unital and associative.
If you're feeling like it, you could instead replace several generators with $\bigvee_H G/H_+$, and establish $G$-spectra as equivalent to modules over one ring spectrum which is a big "matrix algebra" containing a bunch of commuting idempotents. It's not clear to me whether this is generally profitable. (It certainly doesn't make looking at the symmetric monoidal structure on $G$-spectra easier.)
To go further, we need to specify a little about which category of $G$-spectra you're interested in. This is often phrased in terms of a choice of universe.
At one end, you have $G$-spectra indexed on the trivial universe, which are formed by taking the category of $G$-spaces and inverting the suspension functor. There's a "coalescing" result of Elmendorf (his "Systems of fixed point sets") that essentially shows that the homotopy category of $G$-spaces is equivalent to the homotopy category of functors from the orbit category of $G$ to spaces; $G$-spectra indexed on the trivial universe satisfy a similar result.
At the other end, you have $G$-spectra indexed on a complete universe, where all the spheres based on orthogonal representations of $G$ become invertible. These are more complicated, because they're generated by more than just actions $g: Y^H \to Y^{gHg^{-1}}$ and restrictions $Y^K \to Y^H$ for $H < K$. They also have transfer maps.
If you've done any looking into $G$-spectra, you've probably encountered the notion of a Mackey functor, which is a collection of abelian groups with restriction, transfer, and conjugation maps. One compact way to phrase this is that Mackey functors are additive functors from the "Burnside category" to the category of abelian groups. $G$-spectra indexed on a complete universe satisfy a similar result: they are enriched functors from a topological Burnside category to the category of spectra. In particular, every $G$-spectrum produces a Mackey functor to the stable homotopy category. (There are several places I could insert some more or less gratuitous $\infty$-category theory here.)
I know that Clark Barwick has given several talks on this, and is likely in the process of writing it up.
Whether Mackey functors make you happy might depend on whether you're in that pleasant zone between understanding their definitions and trying to do serious homological algebra with them. While I'm writing pithy asides, it's kind of depressing that the Burnside category doesn't have entries on Wikipedia or the nLab for me to link to, and Mackey functors only have this. Many of the presentations in the literature are worth looking at.
There are several categories of $G$-spectra in between, and I don't know much about the general properties of those.
Consider the periodic complex $K$-theory spectrum $KU$. The integral homology group $H_i(KU)$, the direct limit of
$$\dots \to H_{2n+i}(BU)\to H_{2n+2+i}(BU)\to\dots,$$
is a one-dimensional rational vector space if $i$ is even and trivial if $i$ is odd. It follows that $H^1(KU)$ is nontrivial. (It's $Ext(\mathbb Q,\mathbb Z)$.) But this can't be detected in the cohomology of suspension spectra, because $H^{2n+1}(BU)$ is trivial.
So that's an example of a "hyperphantom" map from $KU$ to the Eilenberg-MacLane spectrum $\Sigma H\mathbb Z$.
Best Answer
For any $k$ and any $0<n<\infty$, $K(n)\wedge \pi_{\leq k}S=0$. Indeed, this is true for any spectrum with finitely many homotopy groups, since $K(n)\wedge H\mathbb{Z}=0$ and any such spectrum has a finite filtration into Eilenberg-MacLane spectra. If a spectrum $M$ admits a $\pi_{\leq k}S$-module structure, then $M$ is a retract of $\pi_{\leq k}S\wedge M$, and so $K(n)\wedge M$ is a retract of $K(n)\wedge\pi_{\leq k}S\wedge M=0$ and hence $K(n)\wedge M=0$. But this is not true for $n=1$ and all of your examples (for $MSO$, you must work at an odd prime).