If we take the Churchill speech as the beginning of the Cold War, then the Universal Declaration of Human Rights represents the US putting forth a position in opposition to the Soviet Union.
The 1936 Constitution sounds good, especially beginning at article 118. It gives one a warm feeling like peeing in your pants, but of course Stalin never let the people change their underwear. The intellectuals of the west never had to live under it and therefore revered it, even though the provisions were never followed. How tragic U.S. politicians behaved so inconsistently at the UN that eventually even the Libya of Muammar Gaddafi could reside on the Human Rights Council.
Take note that all the “rights” listed in the USSR constitution (including the successor from 1977) are “positive”, i.e. given by the government (and taken away/nullified at will), rather than, as Obama disdainfully called the Constitution’s Bill of Rights, “negative liberties” that say what the government can never do to you because of those rights being inherent to everyone living.
The UDHR is not in opposition to the USSR. The slippery slope starts at its Article 6. It also recognizes “dissolution” of marriage (divorce) as a right in Article 16, and is not clear that the family is one man married to one woman but uses ambiguous language. Article 19 fails to say “freedom of speech”. Article 23 reads just like the USSR’s “right to work” constitutional articles, including “equal pay for equal work” et al. Article 26 also stipulates free and compulsory education for children, which is stipulated in the Tenth Plank of Communism in Chapter 2 of the Manifesto.
The behavior of US politicians at the UN (after all, it was leftist US politicians that set it up) is not inconsistent with the structure of the UN.