Don't know why the stand is thicker one side, have always been curios' but I was told by a DR To always tie a short rope or the strap supplied in the tool kit back to the frame because they sometimes move when getting the front wheel out, but when I have used the stand in the past without using a strap, I have never had a problem, Was it early elf an safety, ? Andrew.h.
I think Michiel's theory is interesting. In the mid 1930's the bottom stay had special barrel shaped pieces attached on both sides to perform exactly the function Michiel is describing. Those pieces rested against the forks to act as a stop.
However pre war M series bikes didn't have those and didn't have the strenghtened stay either. My 1940 WD M20 still has the "civilian" bottom stay without the strenghtening piece en so does my 1939 WD M20.