Just wanted to say that most people who study birdsong and bird calls wouldn't describe it as a "language"--as far as we know, humans are the only animals with language, unless you believe that one guy who wrote the article about dolphins. But you're right that birdsong is a lot more like language than most other animal vocalizations; often the songs have "syntax" (like in the Weiss article you posted) where the structure follows rules. But the syntax doesn't convey meaning in the same way it does in our language. No bird has ever written a philosophy book, right? They're just communicating stuff like "this is my tree and I'm so manly I can sing in this tree for hours in the middle of the night".
Along the same lines, I think birds are pretty close to us evolutionarily speaking. You're not the first person I've heard compare them to aliens, but their brains are actually a lot like ours. Here's a grumpy blog post I wrote about it if you want to have a look: http://www.theneuroethicsblog.com/2012/09/snakes-on-brain-or...
As the article above demonstrates, along with many other behavioral studies like those you mentioned, birds behave a lot like we do. So: their brains are a lot like ours and their behaviors are a lot like ours. This is why we should study them even more, if we want to understand intelligence.
Think of it this way: you're an alien and you want to figure out how a computer works. If you could only carry out your experiments with MacBooks, it would probably take you longer than if you could also use e.g. Lenovo ThinkPads (just trying to draw an analogy, I'm not throwing shade at Mac or anything).
I hope that wasn't way too "tl;dr" but I just wanted to build a little bit on your comments.