I also watch a lot of old movies, and yes, the camera placement and immobility certainly had MUCH to do with it. It was quite an ordeal to change camera positions, lighting, etc. to accommodate people stepping off sidewalks, walking around the vehicle and getting in the driver side.
Vincente Minnelli is said to have invented a studio crab dolly in the 1940s in order to easily move the camera around under very ideal conditions. But neither the devices nor the cameras were advanced enough for a good number of years to be use in a truly mobile fashion in uncontrolled environments as they are today.
And you may find it interesting that during driver's education, I was taught by my driving instructor (Mr. Perry) that on busy streets, I should use the passenger side to avoid opening doors that might impede oncoming traffic. (Providing of course I was not accompanied by a woman for whom I would be opening the passenger door for as a courtesy) So there was perhaps those influences as well. Although I cannot recall ever seeing my parents or any other adults doing such a thing.
But I suspect the phenomenon of the 'passenger side entry' in old pictures had much more to do with the camera placement, focus, lighting, hitting of marks, and controlling precisely what the camera was to 'see'. The business that occurs within the frame of the cameras in motion pictures of those days was VERY tightly managed by directors! They knew exactly what they wanted seen by the audience as points of focus, and what was to be seen peripherally to fill out the visual experience. The use of computer generated graphics and truly mobile (and sophisticated) cameras have certainly changed those techniques in modern picture making!
Regards, Pete