I am not a lawyer. Not even a bush lawyer. But AFAIK, the IRB is the world governing body, and the ARU is affiliated with them. So if any club or other entity wants to play rugby in this country, the ARU or its nominees would presumably have to consent.
It kind of stands to reason that any serious sport has a formal governance regime.
Broadcasters, governments, insurance companies, etc etc. You name it. They would not be interested in supporting a rebel sport, which is what it would have to be.
I am a lawyer and the idea that you can't play rugby, or set up an organization to play rugby, without IRB approval sounds pretty far fetched. We havn't entered into some kind of multi-lateral international treaty dictating how and when rugby must be played, or who controls it.
I suspect the IRB has authority because of individual agreements with national unions. It's mutually beneficial for a thousand reasons. Standardized rules, World Cup, etc.
The problem is, as you point out, A rival organization would not have access to the rest of the rugby world. Though, as with anything, if it became popular broadcasters and sponsors would get on board.
Note: i have done absolutely no research to back this up.