> the constant sheaf Z is NOT a homotopy sheaf (even though it is an ordinary sheaf)

This type of phrasing is ambiguous and is probably responsible for the confusion.
In this sentence, Z is used to refer to two completely different presheaves:

* the presheaf Z of abelian groups, which sends U to the set of locally constant Z-valued functions on U;

* the presheaf Z[0] of unbounded chain complexes, which sends U to the unbounded chain complex concentrated in degree 0, where it is given by the abelian group of locally constant Z-valued functions on U.

The presheaf Z is indeed a 1-sheaf and an ∞-sheaf of abelian groups.

The presheaf Z[0] is a 1-sheaf of unbounded chain complexes.
It is not an ∞-sheaf of unbounded chain complexes
and its ∞-sheafification can be computed as the ∞-sheaf of integral singular cochains.