The Brooks Brothers history, Generations of Style, explains that shirts with button-down collars, introduced around 1900, became enormously popular during the 1920s. A new generation wanted to set itself apart from the late Victorians and Edwardians who wore stiff, stand-up, detachable shirt collars. The button-down collar was soft, relaxed, even slouchy. The button-down collar's roll moved and shifted as the wearer moved. This makes sense, because supposedly the style originated on shirts worn by polo players (although there is a school of thought disputing that story). Anyway, the movement fit right in with flapper zeitgeist. Think of bobbed hair and dresses with long strings of beads, or the charleston compared to the waltz. Stiffness was out.
This brings us to why I don't like a lining in the collar of a button-down shirt. Anything tending to stiffen the shirt collar and make it look too perfect and immobile goes against the sporty spirit of the button-down. It's supposed to be a soft and slouchy collar, not a stiff and proper collar.
Edit: I should qualify this. Some shirting fabrics, such as broadcloths, are so thin that they do need a lining in a button-down collar. It's button-down oxford cloth shirts that I prefer with unlined collars.