I am confused about the proofs I found of Leray's famous theorem:
Theorem Let $X$ be a topological space, $\mathscr{F}$ a sheaf on $X$ and $\mathscr{U} = \{ U_i \}_{i \in I}$ an open cover of $X$. If $\mathrm{H}^p(V, \mathscr{F}) = 0$ for all $p$ and all $V = U_{i_0}\cap \dots U_{i_n}, i_j \in I$ then $\check{\mathrm{H}}^p(\mathscr{U}, \mathscr{F}) \simeq \mathrm{H}^p(X, \mathscr{F})$ for all $p$.
More specifically, I would like to understand what's wrong about the following proof (which seems to be wrong since it's neither used in [V] = Voisin, Hodge Theory and Algebraic Geometry I, Theorem 4.41 nor [H] = Hartshorne, Algebraic Geometry, Theorem 4.5.)
Proof Let $\mathscr{C}^p = \mathscr{C}^p(\mathscr{U}, \mathscr{F})$ denote the sheafified Cech complex associated to the cover. It is true for any cover that $$0 \to \mathscr{F} \to \mathscr{C}^0 \to \mathscr{C}^1 \to \cdots$$ is exact, or in other words, is a resolution of $\mathscr{F}$. The condition $\check{\mathrm{H}}^p(\mathscr{U}, \mathscr{F}) \simeq \mathrm{H}^p(X, \mathscr{F})$ implies (and is actually equivalent to) the fact that the sheaves $\mathscr{C}^p$ are acyclic with respect to the functor of global sections. Hence $0 \to \mathscr{F} \to \mathscr{C}^\bullet$ is an acyclic resolution of $\mathscr{F}$. Hence $\mathrm{H}^p(X, \mathscr{F})$ can be computed as the cohomology of the complex $\Gamma(\mathscr{C}^\bullet)$. Finally, $\Gamma(\mathscr{C}^\bullet)$ is equal to the (unsheafified) Cech complex for the given cover and hence its cohomology is equal to $\check{\mathrm{H}}^p(\mathscr{U}, \mathscr{F})$.
In both [V] and [H] (I also checked a few others) they use more complicated proofs involving double complexes or induction arguments. So there must be something wrong with my argument, can anyone help me?
The only subtlety I could come up with is that in order to see that $\mathscr{C}^p$ is acyclic I secretly use $$H^p(U,\mathscr{F}|_U) = H^p(X, i_*(\mathscr{F}|_U)),$$ where $i$ is the inclusion map $U \subset X$. However, this seems to be straightforward to prove. For example, take an injective resolution of $\mathscr{F}|_U$ on $U$, push it forward along $i_*$. It is still exact and at least flasque, so it computes the cohomology of $i_*(\mathscr{F}|_U)$.