Why was the atomic bomb not used in the proxy wars of the Cold War?

Many questions have been asked since the use of the atomic bomb in World War II, one being why they were not used during the Cold War particularly during the proxy wars. The Korean War is perhaps a good case in point. So whilst America had lost their nuclear monopoly with the Soviets testing their first atomic bomb in August 29, 1949. In effect however while America had lost the monopoly, at this stage they still enjoyed a nuclear advantage both in number and in delivery systems compared to the USSR. Yet Korea remained a conventional war.

So why were nuclear weapons not used? The atomic bomb was a strategic weapon, a weapon thought to be best used against industrial concentrations, as used in World War II. The Korean War was not this type of war. Arguably, this is why America would go on to develop small tactical nuclear weapons, which could be fired from an artillery piece and effect a much smaller area. The nuclear arms race likewise accounts for America’s unwillingness to use nuclear weapons. The United States had used nuclear weapons, such as Little Boy and Fat Man because they had the advantage and America retained this advantage. For instance in 1953, Eisenhower used nuclear black mail to end the Korean War. Threatening nuclear attack if North Korea and the China refused to negotiate. Consequently, China and North Korea signed the armistice. Arguably, America could now achieve her aims without resorting to a nuclear attack. It is this ability to threaten which makes nuclear weapons powerful. Therefore, nuclear weapons helped start the Cold War and they prevented direct conflict between the super powers.

This is my first, so be kind. Thanks.