For one, it prevented the world from being totally polarized into two opposites which was, (no dear I do not have a mathematical proof for this statement), a good reason for shelling not happening across every border on this planet - well, simply because the idea of if you are not a friend you are an enemy was absent and NAM was the reason.
AFAIK NAM or India was not at all involved in any direct intervention of any war or war like situation (esp. b/w the big two). India was not capable of doing this either with military power or diplomacy. Hell, it just wanted to stay out of the cross-fire having recently been freed and trying to be on its own feet with a plethora or problems at hand related to caste, religion, education and poverty etc.
>>three events that come to mind where the cold war was nearly going hot
Yeah, that is what happens when sensational journalism and films take over. All we see and remember of a decades long conflict are cannons being fired and spies shooting with silencered pistols :-)