Summing up: Whenever you want to "go homotopical" on some category C, a good first step is to embed it into simplicial presheaves, then localize the model structure according to what weak equivalences you want to introduce in C. This is exactly what happens in $A^1$-homotopy theory, for example. To illustrate the difference between the universal properties stated in the first and in the second paragraph: Morel/Voevodsky use the injective model structure to start with, then localize by the $A^1$-equivalences. This is fine, as the injective and the projective model structures are Quillen equivalent and thus represent the same homotopy theory, so they do actually start with the initial homotopy theory containing schemes.An advantage of taking the projective model structure instead (which is also perfectly possible) would be that you get Quillen adjunctions induced easily. E.g. the "complex points" functor from schemes to topological spaces induces a Quillen adjunction from simplicial presheaves with the $A^1$-model structure to the model category of topological spaces which is interesting to study; passing to the homotopy categories it allows you to associate to an $A^1$-homotopy type a topological homotopy type. Some theorems from usual homotopy theory can be recovered from their $A^1$-analoga this way.