Jesus mentioned some stuff about peace, loving your neighbor, taking care of one another, but, yet, he also said to kill those who don't want him to reign over him and he is part of the dramatis personae that is very authoritarian, intolerant, blood thirsty, misogynist, and he is included in the mythology of those who condone slavery and tell slaves to serve. Ergo, even though there is some good, the Bible is not suitable as a guide for morality, as so much of what it considers moral and even necessary via divine mandate we do not tolerate (such as killing your rebellious teenager).
The Constitution gave us fine words, but actions speak louder than words. With the Constitution, we were encouraged and given permission to update it, so that each new generation would not be governed by the dead and they could address their own new problems in their own new society unknown to those who lived during the mid-to-late 18th century. And we did make it better by making sure all men have those rights, and not just men but women, and we are granting these rights to other minority groups.
Where there are flaws and shortcomings, we shouldn't pretend they don't exist. Rather, we should acknowledge the mistakes of the past and boldly declare we can and will do better, and rise to a system were truly everyone does have the same rights and equality under the law, rather than clinging onto systems of past inequities that were/are terribly inadequate and require constant updating because the past didn't actually grant rights to all men despite those words saying it did.