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∞ sends maps of these forms in Top to equivalences in Spectra, because Σ∞ does not commute with Ω. For instance, if X is the space S0, then ΣΩX is a point, and ΣΩX → X is clearly not sent to an equivalence by Σ∞. So, you are not going to get such a functor F which Σ∞ factors through.
You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] and ℤ/(p).