Not trying to defend the UN’s document, only noting some differences:
The Universal Declaration of Human Rights is a generic document, and a bit of a “motherhood statement”, which is not country-specific. It is supposed to cover a broad range of “universal” issues, such as: freedoms, justices, international relations, and so on.. promoting utopian vision & aims - also, it was written in 1948, post-WWII.
The US Constitution & the Bill of Rights, OTOH, were written nearly 2 centuries earlier (1789?), in a completely different historical, political and cultural context, very specific to the US, herself.
Hard to compare the two. But, factually, I agree with Livius: “that many countries accept international treaties as superior to their own law. This is, in fact, one of their legal principles, and its what gives the UN and other such bodies so much clout.”