Users with access to only modify a few pages and also in the group Turn Admin On, can see the New Content panel on every page, including those parts of the site they don't have access to. Trying to add things results in Permission Denied error messages.

It would be preferable to hide the panel (anything that would result in Permission Denied) or otherwise disable them until they are on their own editable pages, in the same way that Clipboard and Version Tag panels are hidden when they're empty.