If $\mathbb{S}$ is the sphere spectrum, I would like to show that the generalized homology theory represented by $\mathbb{S}$, when evaluated on the suspension spectrum $\Sigma^\infty X$ of a pointed ("nice"?) topological space $X$, gives the stable homotopy groups of $X$. That is, we claim $$ \big( \mathbb{S}_*(\Sigma^\infty X) \big)_n \simeq \pi^{\mathrm{stable}}_n(X) $$ where the LHS uses the notation in the 1st link.
When I unraveled the definitions (using the nLab definition of the smash product of sequential spectra), this became the claim that $$ \pi^{\mathrm{stable}}_n(X) \simeq \pi_{2k}(S^{k-n}\wedge \Sigma^k X) \simeq \pi_{2k+1}(S^{k+1-n} \wedge \Sigma^k X) $$ for sufficiently large $k$. Here $\wedge$ is the smash product.
Is there a way to show the above holds? Or, is my assumption incorrect that the sphere spectrum should produce the stable homotopy groups of $X$? (Do we need to replace the suspension pre-spectrum of $X$ by its spectrification, as mentioned here?) (Or also, do I have a mistake in my calculations?)
(Edit: removed an earlier incorrect edit per the suggestions in the comments.)