I have owned two male Cockatiels and so feel qualified to answer this question. Male Cockatiels sing and complex and long mating call around sundown every day. It lasts anywhere from 5 to 45 minutes and is full of notes and warbles and sounds we can't produce, let alone hear clearly. They will often start singing with their beaks closed for the less-intense, beginning, parts of the song, then open up and really belt it out for the louder parts. In the Cockatiels' mating call, there is a significant difference in loudness between the beginning, closed-beak singing and the open-beaked singing. Furthermore, I've observed the closed-beak singing is more like throaty warbling...I dare say it's a warm up to the full-on, ear-splitting, instantly-headache-inducing screaming in the later parts of the mating call.

All this to theorize that birds may practice songs with their beaks closed, but they also can sing with their beaks closed and may be doing a kind of vocal warm-up before hitting the crescendo of their songs.

Hope that helps!