The UN completely botched the Food for Oil program in Iraq, with $ millions siphoned off into corruption. The fewer tasks for the UN, the better. ICANN is doing a lot of things the right way, and they are having to create their program as they go along. This is not easy.
Right now, each country has its own TLD to govern. That in itself is reason enough to keep the UN out. Maybe the UN should have its own TLD, .UN. Then they could hand out .com.un, .org.un, etc
The US government paid for the development of the Internet, including the concept of domain names and TLD's. The UN had nothing to do with it.