The statement that colimits commute with products (in each variable) is a special case of the assumption that colimits are universal (namely, that they are preserved by pullback along the map X -> *), which is one of the axioms on your list.

What you describe works exactly the same way in the setting of quasi-categories: if C is a category which admits small colimits, then any cosimplicial object of C determines a pair of adjoint functors relating C to the quasicategory of simplicial spaces. If C is the quasi-category of spaces and your cosimplicial space is contractible in each degree, the resulting functor from simplicial spaces to spaces is given by taking the colimit.

The example that you cite is a mistake: the category of symmetric spectra is not freely powered in the sense of DAG III. While it is possible to prove strictification results in the setting of symmetric spectra, one cannot do so simply by applying the results of DAG III.