Excellent (and huge) question. This has become a key question in teaching history. Generally and broadly, we can split this into two camps. There are the more conservative elements, who insist that history is history; that is, it's just what happened, it's facts and dates, and it's not open to interpretation and ideology. These are the people who resent when the Founding Fathers are in any way impugned, when the character of America is questioned, and when minorities are brought into the discussion. The other camp, as represented by Roland Takai, who attempt to reframe the American narrative, insists that the voices that have been left out of the conversation, namely those of women, minorities, and the working class, are essential. This approach is exemplified by Zinn's A People's History of the United States and, really, any history that puts emphasis on the people rather than the so-called great figures of history.
With that as a preface, I think there is a great deal of evidence for America failing to live up to its ideals, especially the famous one from the Declaration of Independence that all men are created equal. As many now know, Jefferson, for all his idealism and intelligence, owned slaves, which makes the statement problematic. Few of the Founding Fathers, most of whom were wealthy, property-owning white elitists, cared about the rights of minorities, notably African American slaves. For some, this is a flaw in our system, while for more radical thinkers, this was intentional. That is, America is a country created by and for white men. Racism, then, is not an accident but a founding principle. For this line of thought, I would highly recommend Ibram X. Kendi's Stamped from the Beginning, as well as the work of other African American thinkers like Du Bois, Coates, and Gates.
This is a very superficial treatment of an important topic and, aside from the Takai text, I think reading widely on the subject, especially from the voices of those who have been left out of the American narrative (and the American dream), is crucial. Other books worth looking at are The Guarded Gate, El Norte, and All the Real Indians Died Off.