This identifies a real problem but is the wrong solution. "Admin titles" is overkill and would involve a ton of awkward UI that was ususelss for most people.
The problem is that all text widgets without titles are labeled as *Text*. The effect is that even though they each have custom content, they are indistinguishable when they are collapsed.
Contrast this with non-text widgets like 'Categories'. If they have a title it is used, but if not they still have useful generic names. The text widget label is a special problem because people are likely to have many of them in a row, unlike more generic widgets.
The solution is to show the first few words of the widget content in the heading when the widget is closed. The same way the titles show, ensuring that there is always something to identify a text widget (unless its empty). This way the user can get a sense of what's inside, and in all likelyhood be able to distinguish specific widgets.
Showing the start of the content would not cause any major problems and would help a lot of people.
Maybe that should be its own 'idea' though.