It is well know that the $\infty$-category of group-like $E_\infty$-spaces and the $\infty$-category of connective spectra are equivalent, see e.g. May - "$E_\infty$-spaces, group completions and permutative categories" or Lurie - "Higher Algebra", Remark 184.108.40.206
Now the category of $E_\infty$-spaces (here space means simplicial set) carries a model structure as well as the category of spectra. Is there a direct (left) Quillen functor
$E_\infty$-space $\to$ Spectra
whose derived functor restricts to such an equivalence? I have been unable to find a discussion of this in the litertatur. The only thing I can find are indirect functors going through $\Gamma$-spaces or related categories. The Bar-construction which is usually used is not left Quillen (!?).