As I say in my comment, a possibly useful way to think about this is from $L$-functions, and the Sato-Tate distributions for degree 4 $L$-functions.

There is work of Fite, Kedlaya, Rotger, and Sutherland on this.

In weight 1: http://dx.doi.org/10.1112/S0010437X12000279 http://math.mit.edu/~drew/sato-tate-g2-errata.txt

In weight 3: http://arxiv.org/abs/1212.0256

I will concentrate on the second paper (weight 3).

In brief: almost everything (from dimension counts) should have generic $USp_4$ Sato-Tate group, but neither ${\rm Sym}^3 A$ nor $A\otimes B$ has such a group. So the answer to the specific raised question is "no".

In particular, there are a few "standard" (functorial) operations to construct a degree 4 $L$-function (here symplectic, so indeed self-dual).

You listed ${\rm Sym}^3 A$ and $A\otimes B$, which are in Section 6 of the "weight 3" link above.

Another example is $A\oplus B$, which is Section 5 (the $L$-functions are imprimitive here).

They don't mention inductions from $GL_1$ (Grossencharacters) or $GL_2$ (Hilbert modular forms), but those are other "functorial" constructions.

Most of the "smaller" Sato-Tate groups they list come about from eg complex multiplication, or a tensor/sum of "correlated" $L$-functions.

But the main bulk (the "generic" case) should be Siegel modular forms.

My knowledge is not that deep, but I think there are still "lifts" in the SMF class which give "generic" $L$-functions (the full $Sp_4$ Sato-Tate group), and that these are often used in the literature to exhibit examples, but that the "generic" SMF is not a lift (correspondingly, eg some small level modular forms for $GL_2$ are eta-products, they have the "generic" Sato-Tate group, but perhaps are not truly "generic" in the sense one might want).

Examples of SMF lifts include the Saito-Kurokawa and (Miyawaki-)Ikeda lifts (I am no expert, there are likely more I think, I just know these words). EDIT: these are not really relevant, they give imprimitive deg 4 $L$-functions it seems. In fact, the Yoshida lift would be more relevant (it can give both the direct sum and tensor product from the spinor/standard $L$-functions).

From the standpoint of what is expected, the $d$-column on page 10 of the "weight 3" preprint gives the dimension of the associated space, so indeed almost everything is thought to be

$USp_4$ Sato-Tate group (dimension 10)

while the ${\rm Sym}^3$ construction lands you in $D$ (dimension 3),

the tensor product is generically in the normalizer of $U(2)$ (dimension 2),

the direct sum (dimension 6 or 4) is $G_{3,3}$ or $G_{1,3}$, depending on whether you direct sum the same weights (wt 4 modular forms), or first Tate twist (wt 2 plus wt 4)

the induction from $GL_2$ (Hilbert modular forms) is dimension 6, giving the normalizer of $G_{3,3}$ (they do not mention this one, maybe it's the normalizer of $G_{1,3}$ in non-parallel weight $[2,4]$ and with $G_{3,3}$ in parallel weight $[4,4]$, I would have to think).

the induction from $GL_1$ (Grossencharacters) is in the $F_\bullet$ classes (I think, again they don't do this explicitly, and get various $F$-classes as "degenerations" as mentioned next), so dimension 2 (I don't think a full dihedral example occurs over ${\bf Q}$ in the sense they demand, but $F_{ac}$, with its $C_4$ in the fourth column, should occur for cyclic quartic fields).

Everything else (the cyclic and J-cyclic data) is dimension 1 in the appropriate moduli space, as I say, coming about from having "complex multiplication" on the underlying data in one of the above constructions. For instance, tensor a CM elliptic curve with a CM modular form of weight 3, depending on how the CMs line up, you get different behavior, landing in $F$-classes, see 6.15 (the point being, you did the CM-induction before the tensor, but this can be unwound, so that's why induced $F$-classes appear).

Sorry for such a sprawling answer, and indeed it is certainly not from the automorphic perspective you phrased, but I hope it helps.

In brief: almost everything should have generic $USp_4$ Sato-Tate group, but neither ${\rm Sym}^3 A$ nor $A\otimes B$ has such a group. So the answer is no.

docome about from lifting (though not necessarily the lifts you list), as it is one way to exhibit examples. Writing down a generic SMF is harder (AFAIK). You can also look at the Kedlaya (et al) work on Sato-Tate groups in deg 4. Another non-SMF example that is not in your set of lifts is Hilbert modular forms (over GL2/K, but induced to be thought of as GL4/Q, similarly with Grossenchar over a quartic field). $\endgroup$