The avian brain is fairly different from the primate brain though, so if a bird language area were to exist, it wouldn't necessarily look like the human ones.

The avian brain is fairly different from the primate brain though, so if a bird language area were to exist, it wouldn't necessarily look like the human ones. You're also giving birdsong fairly short shrift here. It is definitely not just mimicry--the songs that birds sing are not exact copies of their fathers' songs and the songs' length and complexity plays a role in attracting mates. It's unclear if birds can learn sequences which are as complex as human languages but it's certainly not as settled as you make it sound. It looks like finches do learn regular languages, but the center-embedding stuff is a lot more controversial (and, as a counterpoint, humans don't actually nest center-embedding clauses very deeply either).