re: Why would we need cooperation?-because "Pride cometh before a fall".
In 1950 the Chinese share of global GDP was 4.5% (give or take), now it is about 14%. In contrast, the 1950 US share of global GDP was 27%, now it is about 19% ( and US public debt is roughly 65% of GDP and rising).
In 1913 the share of global GDP by the British Empire was about 21%, by 1950 the Empire had effectively ceased to exist. Times change.
Of course the US could go to the moon without anyone's help, it already has, and it will remain rich and powerful for as long as any of us are alive, but it will soon have to share that privilege. It would be better to have your rivals sweet than sour.