"If not, how would you recommend changing things if you were the Secretary General of the UN?"

The problem is once you go to the UN about space rights you endow it with the legitimacy to make the rules. This is a dangerous assumption. Do we really want the same organisation which has the world's worst human rights offenders sitting on its Human Rights committee to lord it over a on a global space treaty?

I don't think the UN should be involved at all. If anything a new global org should be set up; chaired principally by those nations most active in space at the moment, with more nominal seats for developing nations. They get their rights reserved but until they can act upon space projects they should not be vetoing or obstructing a treaty for property or leasing rights in space.

A UN treaty for space will always settle for a lowest common denominator solution.