This description has very little to do with reality. When it came out in 1940, it was hardly the first anti-Nazi film. Britain was already at war with the Nazis, and the Americans were already fighting them at sea. It is extremely unlikely that Hollywood studios were terribly concerned about selling movies to the Germans considering they would have to cross a blockade and net of submarines to get them there. If by using “his own money” they mean using the production company he helped found, United Artists, then yes, that is true, but it was one of the largest in Hollywood at the time.
The only place the film met with real resistance was South America, where several countries had issues screening it due to Nazi sympathizers.