I have two, somehow related questions, and I would be very grateful if you point out at some references if the answer is known. If these are too elementary or not research level, then please feel free to move it mathexchange. I am not an expert in this, so I thought it is worth asking here.

According to Ravenel's orange book, the Nilpotence Theorem implies that $H\mathbb{F}$ and $K(n)$ are ``essentially'' the only homology theories for which the Kunneth isomorphism holds; I suppose this latter means that $$E_*(X\times Y)\simeq E_*(X)\otimes_{E_*(pt)}E_*(Y).$$ So, my first question is whether this conclusion is immediate or whether it is not that trivial and a proof is recorded somewhere? I have not tried to prove it.

Second, I wonder if there is a spectrum $E$ so that $$\pi_*L_E(S^0\wedge S^0)\simeq\pi_*L_ES^0\otimes\pi_*L_ES^0$$ where the tensor product is over some suitable coefficient ring, $L_E$ is the Bousfield localisation with respect to $E$, and $S^0$ is the sphere spectrum. Or, equivalently on the category of spaces $$\pi^s_*L_E(X\times Y)\simeq\pi^s_*L_EX\otimes\pi^s_*L_EY$$ again with tensor product over say $\pi_*L_ES^0$. A first guess is $E=H\mathbb{F}$ and $K(n)$, but these do not seem that immediate to me either.

**Edit**. Perhaps it is better to replace $\times$ with $\wedge$, and ask for an isomorphism
$$\pi^s_*L_E(X\wedge Y)\simeq\pi^s_*L_EX\otimes_{\pi_*L_ES^0}\pi^s_*L_EY$$
as also in Neil's update to his answer.

The motivating example for the second question is the following. Consider a compact Lie group and an embedding $1^{\times n}<G^{\times n}$. Then, the transfer map of Becker-Schultz-Mann-Miller-Miller provides a transfer map (upon choosing a suitable twisting bundle) $$\Sigma^{n\dim(\mathfrak{g})}BG^{\wedge n}_+\to (B1^{\times n})_+=S^0.$$ Now, one may hope that in homotopy the image of $$\pi_*(\Sigma^{n\dim(\mathfrak{g})}BG^{\wedge n}_+)\to\pi_* B1^{\times n}_+=\pi_*S^0=\pi_*^s$$ falls into the submodule of decomposable elements in $\pi_*^s$. Here, $\mathfrak{g}$ is the Lie algebra of $G$. This, however, is not true, which I think is mainly because of the lack of a Kunneth theorem. So, one hope is to find some $E$ where homotopy of $E$-localised spaces/spectra admits some Kunneth theorem. For this reason, I am interested in a Kunneth theorem and not a Kunneth spectral sequence.

Equivalently, if there is a way to describe the submodule of decomposable elements in $\pi_*^s$ then I would be happy.