Without getting too technical about flying, the older planes (supposed to be an Airco DH.4, or maybe a Curtiss JN-4 but in the movie they used Stearman 75s) had a stall speed of about 35 MPH, That means the ship lost lift and stopped flying at that speed. Additionally the planes of the time (the 1920's) were not very heavy: about 2000 pounds fully loaded. Finally, they were taildraggers, meaning they had the third wheel at the back of the plane, not the front.
All of these elements combine to make a ship that will land in just a couple hundred feet, and with those larger tires and being a taildragger the ground can be pretty rough at that. (Concrete/paved runways were years off in the future: all planes then were designed to take off and land on grass strips.) In fact, landing would not be the issue: short-field landing techniques enable me to land my 2500 pound plane in 800 feet; no, the big concern is getting out again after adding the weight of the 2 new occupants.
The ground near the river looked as though it would easily handle the distance needed to get in/get out.
..Joe