I am not sure they even take reservations if you call them directly. They might take a call-ahead seating. It is true, even though Big River is on the dining plan, it is not a Disney-operated restaurant. Big River is operated by Gordon Biersch Brewery Restaurant Group.
There are several restaurants that don't take reservations. Some of them (like ESPN and Beaches & Cream) are actually Disney-operated restaurants. There are also a number of restaurants that are not operated by Disney that do take ADRs through the Disney reservation system.