This is not quite true, while you're correct that there is no third gate, the Greenway (as pictured in the upper left corner of the 3 picture illustration you'r

This is not quite true, while you're correct that there is no third gate, the Greenway (as pictured in the upper left corner of the 3 picture illustration you're referring to) is not the road which is shown in the "close-up" (lower center picture) of Bree which goes "To Deadman Dike". The illustrations are taken from "The Atlas of Middle-Earth" and the road which goes diagonally north-west from the prancing pony towards the greenway (Old North Road) is described as follows in "The Atlas of Middle-Earth" (pg. 124; At the Prancing Pony): > A drawing shows a lane which curves north from the Great [East] Road, one branch climbing to the crest of the hill, while another leads through a small opening in the hedge for a shorter route to Combe and Archet. The "North Gate" shown on the picture is the "small opening through the hedge". I'd say it was probably marked as a "Gate" here since Bree is surrounded by a Dike/Hedge and should simply illustrate that you can't simply cross on a "normal" road here. The drawing which is being referred to here appears in `The Return of The Shadow` (History of Middle-Earth Book 6) as `Plan of Bree` in Chapter `XX The Third Phase (2): At the Sign of the Prancing Pony` The drawing "Plan of Bree" by Tolkien can also be seen on pg 45 (Illustration 25) in "The Art of The Lord of the Rings"