Although Σ and Ω are autoequivalences of Spectra, so that the maps X → ΩΣX and ΣΩX → X are equivalences, that does not mean that Σ<sup>∞</sup> sends maps of these forms in Top to equivalences in Spectra, because Σ<sup>∞</sup> does not commute with Ω. For instance, if X is the space S<sup>0</sup>, then ΣΩX is a point, and ΣΩX → X is clearly not sent to an equivalence by Σ<sup>∞</sup>. So, you are not going to get such a functor F which Σ<sup>∞</sup> factors through. You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top as a presentable (∞,1)-category. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] and ℤ/(p).