Isn't the point of the NTO to verify one's credentials to be actually on NATO business, in conjunction with one's service ID card? In other words, it validates the use of the service ID for international travel within NATO countries.
I'd guarantee trying to use a MOD90/F1250/Navy whatever at a US POE without the NTO would be fruitless, and result in a) a long trip back and b) difficult questions next time.
As for the "eye scan" and bit of paper to give back, it's a) a photo, and b) a departure record, as it says clearly on it. If you overstay, you are likely to be banned, length of ban varying by length of overstay. Since the departure record is the document that determines whether or not you overstayed, it is a tad important. Of course people genuinely lose them, and don't get screwed, as long as they report it and do the paperwork thing. The airlines won't accept you for travel without one, if you need one.