Mind you as Von Braun was being debriefed and building his red stone missile and as JPL were also completing there missiles there were several others completing theirs as well. Bell labs for instance already had a missile before von braun came to the U.S :
That is how the U.S military complex works. The government then picks the best system from those developed.
So when you say Von Braun taught the Americans it is a falacy because even before he was born the U.S was already researching into rockets.
Von Braun himself studied the work of American Robert Goddard in campus. The Nazis even tried to contact Goddard to get his publications. The V2 was based on Goddard’s work!