Why can't I see the task board for a plan in the web UI?

The different views on the planned items tab of a plan are preconfigured for a specific plan type. To have taskboard view, just add a new view to the plan (i.e. create a view) and on the options select Display -> Taskboard. Click ok on the view creation dialog of the plan and you see it. On the specific link that you posted it is possible to start the addition of the new view, but you will probably not be able to save your work, because you have not the permission on that plan. But you can watch the plan in taskboard view.
Hope that helps.
Henning