Where in literature can one find a construction of Steenrod reduced powers (for an odd $p$) that
(1) works for the singular cohomology of arbitrary topological spaces (or, more generally, for the cohomology of simplicial sets) and
(2) does not use homotopy-theoretic constructions (such as $K(\pi,n)$ spaces)?
I like the approach of the Steenrod--Epstein book. Unfortunately, the main construction is done there for the cellular cohomology of finite regular cell complexes. Then the operations are extended to infinite regular complexes and then to arbitrary CW-complexes (using approximation by a simplicial complex); then this is applied to the singular simplicial set of a topological space. This way seems somewhat indirect if we are interested in singular cohomology.
Another approach is given in Switzer's "Algebraic topology" for $p=2$. It starts with the words "By the acyclic model theorem, there exists a natural equivariant chain map $W\otimes S(X)\to S(X)\otimes S(X)$" (where $S(X)$ is the singular chain complex and $W$ is the free resolution of the group $\mathbb Z/2$). Some other textbooks also use this approach. Unfortunately, I could not find an exposition for $p$ odd.