I know that the Mayer–Vietoris sequence for sheaf cohomology can be derived from the spectral sequence relating Čech/presheaf cohomology to sheaf cohomology, but I am wondering if the following seemingly much simpler construction is correct.
Let $\mathcal F$ be a sheaf on a topological space $X$ and $X=U\cup V$ an open cover. Pick a flasque (e.g. injective) resolution $\mathcal F\to\mathcal I^\bullet$. Then the sheaf sequence gives a short exact sequence of cochain complexes of Abelian groups: $$ 0\to\mathcal I^\bullet(X)\to\mathcal I^\bullet(U)\oplus\mathcal I^\bullet(V)\to\mathcal I^\bullet(U\cap V)\to 0. $$ Exactness of the first three arrows is the sheaf property, and surjectivity at the end comes from flasqueness. Taking cohomology of these cochain complexes thus gives an exact sequence of cohomology groups of the form given by M.–V.
Is there something simple I'm missing that causes this construction to fail? (I suppose I haven't checked well-definedness or agreement with the real M.–V. sequence.) Also it seems that, if correct, it generalizes to covers by more than two opens.