This is not true. For example for $S^1$-spectra there is the descent spectral sequence
$$ H^s_{Nis}(X,\pi_tN) \Rightarrow \pi_{t-s}\mathrm{Map}(\Sigma^∞_+X,N)=\mathrm{Hom}(\Sigma^∞_+X, N[t-s])$$
and the statement you want would be like saying that the $E_2$-page is concentrated in the $t=0$ line.

In fact I do not think that $S_N=\pi_0\mathrm{Map}(\Sigma^∞_+-,N)$ is a Nisnevich sheaf at all. This would be like saying that if a complex of presheaves satisfies hyperdescent, then all of its homology presheaves are sheaves.