If reachability is in $\mbox{PSPACE}$, PSPACE, then it's in $\mbox{NP}$ NP because during the search once you reach the target you can immediately stop, without backtracking results. Unlikely Othello, in which a configuration requires children configurations to return a value, reachability can be done with non-deterministic machine. It doesn't require keeping the list of visited configurations. You can forget them immediately. If you still don't understand what I'm saying, contrast the difference between Othello evaluation and reachability.
P.S. The exponential-length possibility is ruled out since it's in PSPACE. Otherwise PSPACE is not enough to store the current path. Here you need to store the path because PSPACE machines are not non-deterministic (or branching), hence it has to backtrack.
