They are. For example, the ARU can make a determination on anything to do with Australian rugby without SANZAAR input.
The ARU is a constituent of SANZAAR and so it holds a vote in the organization's decisions plus it has a right of veto.
A useful analogy would be the United Nations Security Council. It's member states are clearly distinct entities, and consensus is required for the UNSC to act. Each member also has right to veto.
But these teams have agreements with the ARU regarding a SANZAAR competition. Either these agreements extend to SANZAAR as a whole by default or the ARU have acted improperly in signing them? The whole reason I say the ARU and SANZAAR are effectively the same thing is because of the right of veto.
I just don't see how SANZAAR is able to cut an Australian team if the ARU has no legal right to cut a team, given the ARU have the right within SANZAAR to say 'sorry turns out we can't do this.'
To use your analogy, it'd be like the UN Security Council agreeing on some form of future action, only for a legal challenge in one of the permanent member countries to rule it illegal for the government to agree to it and thus force that government to use their veto (given this would all be before the actual action was taken).