That is an excellent point. The answer is yes, but I'm very fond of minimal complexes. (I'm much less fond of minimal fibrations). So I prefer the illuminating proof that uses them. However, this result is the simplicial analogue of the Whitehead theorem that a weak equivalence between CW complexes is a homotopy equivalence, and it is possible to mimic the proof of that result. After the fact, of course, these are both instances of the model theoretic statement that a weak equivalence between cofibrant and fibrant objects is a homotopy equivalence.

You are mixing contexts and definitions in a fairly confused way. $E_{\infty}$ algebras make sense in many categories. There are many symmetric monoidal categories of spectra (historically, the first, by a nose, was constructed in EKMM (Elmendorf-Kriz-Mandell-May). In any such category, commutative monoids, alias commutative ring spectra, are equivalent to $E_{\infty}$ algebras in the relevant category of spectra.