Why does Coughlin insist on the idea that "we don't have to be communists, nor do we have to be bureaucrats?"in the speech "address to townsendites" in 1936
Coughlin insists on this idea because he sees these groups as threats to American freedom and prosperity.
In this speech, Coughlin is arguing that America is in danger of losing its freedom. He is saying that the politicians in both parties are giving up America's freedom and replacing it with things like plutocracy (rule by the rich), bureaucracy (too many government rules carried out by bureaucrats), and communism. It is difficult to understand how the country could simultaneously be plutocratic and communistic, but this is what Coughlin is saying.
In the passage you mention, then, Coughlin is saying that it does not have to be this way. We can reject what the government and the elites are pushing us towards and simply be free and prosperous Americans by standing up to the elites.