Eye marks are also used for post press equipment after sheetfed offset. As an example, an eyemark may be used to trigger a labeller or used on diecutters with specific register features. It is true that the majority are likely used for web scenarios, but if you really want to know why then simply ask the dieline supplier.
As for why they have to stop as a band is simply due to the fact that if there isn't enough contrast with the backround colour or image it won't trigger the camera or sensor looking for it. You can think of these sensor/cameras like black and white, and a lack of a clear eye mark contrasting the background can be an issue.