I am trying to read the Hovey-Shipley-Smith article as defining the stable model structure on symmetric spectra as a left Bousfield localization (as explained on nLab) of the projective level model structure on symmetric spectra, which has level equivalences as weak equivalences.

Almost all ingredients are there in the article. All I have left to show is that the injective Omega-spectra are indeed the S-local objects, where S is the set of level equivalences. The corollary 3.1.8 that every level equivalence induces a weak equivalence of simplicial hom-sets $Map_{Sp^\Sigma}(f,E)$ for E a injective Omega-spectrum. Conversely, lemma 3.1.5 and example 3.1.10 conspire to tell you that if a symmetric spectrum is S-local and injective, it is an Omega-spectrum. So, what remains: is any S-local symmetric spectrum injective?