The constitution is the document that defines the existence of the United States and all of the rights, freedoms, and protections it provides and aspires to provide. It’s not perfect, but it is a heck of a lot better than blind adherence to the flag and whoever waves it.
Seeing as the constitution defines the United States, I think you get a lot closer to the truth when reflecting on it than you do when worshiping the flag or the military or whatever else it is that July 4th is supposed to be about.
It is also important to remember that the constitution is a living document, meant to grow and evolve over time. If there are problems - and there are - it is up to us to change it.