I'm not sure if the card even says "United States," so I'm kind of at a loss of how you can say that USFG = people, and I don't think this argument is that bad.
Regardless of whether the United States is a democracy, you've made a card that claims "Democratic Government = The People," and if that's the interp that you want to read then go ahead. I just think that the card doesn't make the claim that you think it makes. I think even saying a Democratic Government = the People is a stretch because all this card really says is that "Democracy = the people".
I'm not really sure if the author says that America's government is similar to other governments because it says that "These precepts are embodied in the Universal Declaration of Human Rights and in modern constitutions the world over." which doesn't really say anything of how our constitution is a model for other countries.