Here is an answer to a slightly different question, namely, what can one do once it is established that projective sheaves very often do not exist. For a nonaffine scheme, there is no known analogue of projective quasi-coherent sheaves on an affine scheme, but there is an analogue of the unbounded homotopy category of complexes of projectives.

Namely, Amnon Neeman has proven that the homotopy category of complexes of projective modules over an (arbitrary, noncommutative) ring is equivalent to the quotient category of the homotopy category of complexes of flat modules by the triangulated subcategory of complexes of acyclic complexes of flat modules with flat modules of cocycles. Building upon this result, Daniel Murfet in his Ph.D. thesis studies the mock homotopy category of projectives on a separated Noetherian scheme, defined as the quotient category of the homotopy category of unbounded complexes of flat quasi-coherent sheaves by the triangulated subcategory of pure acyclic complexes.

Here is an answer to a slightly different question, namely, what can one do once it is established that projective sheaves very often do not exist. For a nonaffine scheme, there is no known analogue of projective quasi-coherent sheaves on an affine scheme, but there is an analogue of the unbounded homotopy category of complexes of projectives.

Namely, Amnon Neeman has proven that the homotopy category of complexes of projective modules over an (arbitrary, noncommutative) ring is equivalent to the quotient category of the homotopy category of complexes of flat modules by the triangulated subcategory of acyclic complexes of flat modules with flat modules of cocycles. Building upon this result, Daniel Murfet in his Ph.D. thesis studies the mock homotopy category of projectives on a separated Noetherian scheme, defined as the quotient category of the homotopy category of unbounded complexes of flat quasi-coherent sheaves by the triangulated subcategory of pure acyclic complexes.