Performing Transitive Reduction via neighbourhood and strongly connected components
The standard definition of the transitive reduction of a graph $G$ says that we delete as many edges as possible, to retain the same reachability relationships.  However, the vertex set is unchanged: no vertices may be deleted.

The procedure you describe appears to involve deleting vertices.  Therefore, it does not appear to match the standard definition of transitive reduction.  (Perhaps your book has a non-standard definition of transitive reduction; if so, you might need to edit your question to ask about it.)

That said, a related construction does appear to correctly provide the transitive reduction.  In the related construction, instead of collapsing all vertices in a SCC into a single vertex, we replace the SCC with a directed cycle; then we apply transitive reduction to the dag of SCCs.  It is possible that your textbook actually uses the following procedure: (1) shrink each SCC to a single vertex, (2) apply transitive reduction to the resulting graph (which is the dag of SCCs), (3) now take each shrunken vertex and restore all of the original vertices and turn it into a directed cycle.  That should provide a correct algorithm for transitive reduction.  If that is what the book had in mind, your mistake was that you failed to perform step (3): you performed steps (1) and (2) and stopped there.  Of course, you have to perform the full procedure to get the full transitive reduction.

You might also be interested in the notion of a minimum equivalent graph, which involves deleting both edges and vertices; however, finding it in general is NP-hard.