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.
To summarize, regardless of the site (manifolds or schemes), presheaves of connective objects (i.e., with homotopy groups concentrated in nonnegative degrees), whether sets, abelian groups, simplicial sets, nonnegatively graded chain complexes, or connective spectra, are automatically ∞-sheaves, provided their individual values have vanishing homotopy groups in degree 1 and above, and their π_0 is a sheaf in the ordinary sense.
On the other hand, passing to the nonconnective setting, whether unbounded chain complexes or spectra, almost always destroys the ∞-sheaf property, precisely because higher sheaf cohomology groups can be nonvanishing. (Exceptions exist in special cases, e.g., quasicoherent sheaves on affine schemes or Stein spaces, or representable sheaves of abelian Lie groups on cartesian spaces.)
Indeed, the inclusion of connective objects into nonconnective objects already fails to preserve (say) pullbacks. For example, consider an object A with trivial π_k for all k≥1. (Assume all objects are pointed, for simplicity.) The loop space of A is the homotopy pullback of 1→A←1. In the nonconnective setting, π_{−1} of this homotopy pullback will be nontrivial, whereas in the connective setting the homotopy pullback is 1.