This answer takes a different point of view to that expressed in the body of your question, but is relevant to the question in the title.

A sheaf is called soft if any section over a closed subset of $X$ extends to a section over $X$. As the linked wikipedia article states, on a paracompact Hausdorff space, soft implies acyclic. The typical roots of unity arguments in differential topology can be interpreted as using showing that sheafs of smooth functions, smooth sections of bundles, and so on, are soft. (One can always extend smooth functions locally from a closed set to a neighbourhood, and the partitions of unity allow one to patch these extensions.) Another way to phrase this is that fine implies soft .

This answer takes a different point of view to that expressed in the body of your question, but is relevant to the question in the title.

A sheaf is called soft if any section over a closed subset of $X$ extends to a section over $X$. As the linked wikipedia article states, on a paracompact Hausdorff space, soft implies acyclic. The typical partitions of unity arguments in differential topology can be interpreted as using showing that sheafs of smooth functions, smooth sections of bundles, and so on, are soft. (One can always extend smooth functions locally from a closed set to a neighbourhood, and the partitions of unity allow one to patch these extensions.) Another way to phrase this is that fine implies soft .