There is still much debate about why the Soviet Union – which was consistently way ahead in the early days of space exploration, failed to beat the USA to putting a man on the Moon. But while there is some disagreement over which factors were the most important, there is considerable consensus about which factors drove this.
They started later.
The USA made putting a man on the Moon the key national objective, from before they had even put a man in orbit. Pretty much the entire space program focused on this objective. By the time this became a national objective in the Soviet Union, 2 years later, time was very tight to develop a powerful enough rocket, and get the required expertise in flight systems.
Also, at the time the speech was made, the Soviet Union was so obviously far ahead, they did not take the US intention seriously.
The NK-33 engines were originally built for the Soviet Moon Rocket, the N-1. (Under the designation of NK-15, and NK-15V for the high altitude version). This design was a direct result of a blazing row between the Chief Designer, (Sergey Korolev), and the best rocket engine designer, Valentin Glushko. Glushko wanted to use propellants which Korolov considered far too dangerous. So Korolev turned to Nikolai Kuznetsov, who up until that point had only designed engines for jet aircraft. Large rocket engines are notoriously difficult to design, due to combustion instability, so they were pretty much forced into a large number of smaller engines.
Many consider this a key reason for the failure of the N-1 program.