How to prove Leray's theorem without using Sheaf cohomology

Question

Let $X$ be a manifold. Leray's Theorem for Čech Cohomology states that if a covering $\mathfrak{U}$ is acyclic $(\check{H}^p(U_{i_0}\cap\ldots U_{i_k},F)=0$ for every finite intersections of elements in $\mathfrak{U}$ and every $p$) for a sheaf $F$, then $$ \check{H}^q(\mathfrak{U},F)=\check{H}^q(X,F) $$ where the first group is the Čech cohomology group depending of the covering and the second one is the Cech cohomology group independent of the covering (taking direct limits by refinements). Every proof I have seen uses sheaf cohomology to compare $\check{H}^q(\mathfrak{U},F)$ with the sheaf cohomology group $H^q(X,F)$ and then conclude that it agrees with $\check{H}^q(X,F)$. Is there any way to prove it without using the strong machinery of sheaf cohomology? It is indeed a theorem in Čech Cohomology theory so my intuition is that we could avoid sheaf cohomology theory. Any references for (pure) Čech cohomology (proving theorems without using equivalences with other cohomology theories) are welcome.