Such sequence of localizations/colocalizations does not exist, since homotopy is not concrete.
Suppose, for contradiction, that such sequence is constructed, then we obtain a corresponding sequence of reflections/coreflections on the level of homotopy categories, producing a fully faithful embedding of $\mathrm{Ho}(sSet_{\mathrm{standard}})\rightarrow \mathrm{Ho}(sSet_{\mathrm{trivial}})\cong sSet$. Next, it is possible to construct a faithful functor $sSet\rightarrow Set$, say, by sending every simplicial set into the product of the sets of its simplices, obtaining a contradiction with Freyd's theorem.