The only way that I can delete the text is on the Layout view, or directly on InDesign.
When you open an assignment file, you'll only be able to edit text in frames that are 1) listed in the Assignments panel; and 2) that you've checked out to yourself. Even in Story/Galley view, if your cursor is blinking in a story that you've checked out to yourself, you'll see "Editing" appear in the grey story bar at the beginning of that story, and the story will be highlighted in the Assignments panel, and there it will show the Pencil icon.
In stories that are part of an assignment but that you haven't checked out, you can select text and copy it, but you can't edit/delete text (in any view).
AM