"If you do that, you have mitigated your "Duty to warn" to the extent humanly possible".
No this is far from the facts. Yes, you may have met the burden of "Duty to Warn", which is great, BUT, you still have a responsibility to ensure that your pet can not do harm to someone else. The ONLY way that can be done is that the bird would have to be restrained when people visit. It's like having your dog in your yard with a sign nailed to a tree "vicious dog", but then the dog bites the mailman or mail person when they enter the yard to deliver the mail. But if the yard is fenced and the sign posted to the front of the fence would definitely mitigate responsibility.
Anyone can sue anyone over anything. That is a given. Whether that suit is successful or not is another matter.
Any animal that MAY bite and cause damage is legally the owner's responsibility. That includes pretty much every large parrot, out there, and quite a few of the little ones... And since they ALL have the propensity to bite, we have a legal duty to warn.
I believe keeping a wild animal in a domesticated setting only INCREASES liability for bites. It's not really a valid legal argument, that hey, technically, it's a wild animal, and the idiot stuck his finger in the cage. "Well then why did you have it..."
But, if the birds are clipped, and kept in their own room, and the sign is posted, and someone APPROACHES THEM and something happens, or they do something stupid... and get bit. Sorry... You ignored the sign and approached them...
Just like the idiot who wanted to pet the pretty tiger at Marine World because he saw a trainer do it. Ummm...the tiger knows the handler, who is a trained professional. You don't belong there. Marine World still gets popped for not securing the area better so that the idiot couldn't get in there, but the idiot who got bit is largely responsible for his own mauling.
And that's the difference.
We are going to be contributorily responsible even if someone lets their little kid stick their finger in the cage when we aren't looking. That's reality!
However, if you do your best to mitigate it, your responsibility, and your insurance rates, will go down. (Well, maybe not your insurance rates...)