I don't know about any work on closed-beak singing as such, not being anybody's ideal ornithologist, but as for rehearsing, well... practising, it certainly happens in various forms and contexts, depending on thespecies of birds and their contexts.

For one thing, many or most birds that vocalise to any high degree (ie most birds) are not born with their song complete; to develop it fully they need to listen to their parents or peers or both. If they are deprived of such vocal role models their song develops poorly and atypically. What is more, local populations of the same species often learn different songs and notes, developing locla dialects that savvy ornithologists can use to place a bird from a familiar area as confidently as it once was possible to place a cockney as having grown up in a particular district. What is more, often a bird that was raised by species other than that of its biological parents often adopt either their song, or thei dialect or both, or at least has its song influenced by their calls.

In South Africa at least we get antiphonally singing shrikes (and/or "bush-shrikes)  and robins (or "robin-chats", depending on who is talking).  The species best known to me is the Bokmakierie (Telophorus zeylonus). A couple will sing in harmony, distinguishing themselves from interlopers that sing in different notes  If you are lucky enough to have a nest of bokmakieries fledging in your garden, you may be treated for a week or two to the family choruses in which the youngsters try out their songs in turn or harmony in delightful practice sessions of clucks and liquid whistles and rattles.

It is by far the most enjoyable birdsong I have experienced, but it is a rare treat.

Anyway, the sum is that birds practice in various and often complex ways and for various purposes, including identification, territorial and courting.

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!

.. in a nearby tree, who was singing very quietly without opening its beak.

That sounds like avis ventriloquisticus   :=)

Georg