Here's a simple example of a Reedy category $R$ for which the inclusion $R^+ \to R$ is not initial. Take $R$ to be the Reedy category freely generated by a single degree-decreasing map $1 \to 0$. The only degree-increasing maps are the identities, so $R^+ \to R$ is the object-set inclusion $\{0,1\} \to R$, which is not a weak homotopy equivalence, and hence not an initial functor.