The broadest consequential public misunderstanding about the U.S. Constitution resembles common misunderstandings about nearly all historical phenomena: context. Everything has a history. Nothing can be properly understood, especially from a policy perspective, without an understanding of the historical (and sometimes geographical) context within which it occurred.
Where one stands on the political spectrum or the range of constitutional interpretation, it’s impossible to understand the text without context. What did “We the People” mean in 1787, when some people were legally human property? Within the context of Anglo-American government experience at that moment, what did “advise and consent” mean?