Finally figured out what was wrong. The data folder had incorrect
permissions after unzipping the base backup. For me, the solution was
unchecking the "Inherit from parent the permission entries that apply to
child objects" option in the Advanced Security Settings dialog for the
data folder & giving the postgres user full control.
Nothing appeared in the log when the database failed to startup b/c the
permissions were wrong. However, an application error did get reported
to Windows which I found using the Event Viewer.
--Rob