There is no "official" method of getting people out. The Birdcage is designed so that things go in but nothing ever leaves. There's exactly one way of leaving in canon, and that's Doormaker and Clairvoyant.
What are you talking about? Governments love black sites where they can throw people and never hear from them again. It's the public that cares about human rights and so on, and as far as the public knows, it's literally impossible to ever leave the Birdcage.
There being a secret way out of the Birdcage doesn't help justify the prison because the people who care don't know it exists.
Legally a prison with no way out violates the US constitution, so they would need to at least pretend such an exit actually existed.
Obviously the government didn't actually think said portals did exist, it was just an excuse for the public.
I always found this idea one of the more bafflingly stupid ones in canon.
In this case, the "inescapable" prison still
needs a release method, and it must be known. Not to the public, but to the government.
Regardless of what form that may take, the simple questions needs to be asked and answered.
1. What if someone innocent gets thrown in.
This is the legal thing, otherwise it would not be allowed to exist by U.S. Law.
2. What if someone
important gets thrown in and is needed? Like, say, a foreign leader gets tossed in and they need to release him to avoid a war?
3. What if the Birdcage becomes
compromised and people are thrown in, like say the President? They
need to get them out, and the U.S. would not stand for that option not existing.
They would have to know and verify that the release method works, because very few countries would stand for it otherwise. Even if they have no intention of using it, it
needs to exist.