The point is that international states become legitimate when they have international recognition. That's really not even 100% based on the United Nations.
When the Soviet Union came apart, it was individual countries.... For example, Canada was one of the first to recognize a number of the countries involved, such as Ukraine and so on. It's about the actual states' recognition.
The United Nations is an umbrella organization, but it is not a world government. It's the actions of individual states that count. Individual states have recognized certain countries, and that's what counts.