Depends how long the cursor is alive. Url containing ?date=lt:2012-12-27 is stateless, user readable and more likely to outlive ?cursorId=xyaxjeasu&page=2.
Now it might not be as accurate as having a cursor but it is a lot easier to implement, and still beats simply ?page=2 without cursor, which is the level you see in most apis today.
The cursor is never invalidated. You can always get a valid response from any cursor, whether the resource exists or was already deleted. That's the whole point of using a cursor.
Perhaps your confusion lies in assuming that a cursor's life cycle is tied to a specific object. It isn't. A cursor means "get me whatever resources would immediately follow this resource". It matters nothing if the cursor exists or not. When you run the query, you will get exactly the resource which would follow the cursor. That's by design.
Now it might not be as accurate as having a cursor but it is a lot easier to implement, and still beats simply ?page=2 without cursor, which is the level you see in most apis today.