The standard (" transverse") flute has pretty terrible ergonomics: your arms are way out to the side. Most woodwinds use a much more natural playing position where the arms are comfortably in front of you, including some (" end-blown") flutes. In the case of the flute, however, this gives a pretty different sound. Could we give up on the simple cylindrical design of the flute to get some thing that sounds like a transverse flute and plays like a transverse flute but is more ergonomic?
Yes! By putting a bend in the head joint it's possible to do just that. This is a common configuration for bass flutes and lower:
Jupiter
JBF1100E bass flute
Pearl PFC-905 contrabass flute
You can get a replacement head joint for a standard concert flute, though it's pricy:
I don't know why this isn't more common? Do people not like that it looks unusual? Katherine Saenger looked into this some (pdf) and claims that the main options for concert flute today don't sound great but better-sounding ones are possible. Has anyone tried these?

How would you efficiently damage the hearing of the player next to you if you weren't projecting most of your sound straight into their ear? (mostly joking)
Seriously though, at least in low brass, the direction that the bell points has an impact on how an instrument sounds to the audience. Tubas designed for use indoors tend to have bells which point upward, whereas tubas and sousas designed for use outdoors tend to point toward the audience. I would not be at all surprised to discover that flute tone is similarly variable based on whether you're pointing the thing at the room or at the floor.
I suspect that your question has a simpler answer, though, which you allude to in your post: The special heads are rare and expensive. They're uncommon, pricey, and do not clearly convey an immediate advantage. There's probably a learning curve to them, as Oural mentioned -- not only will the player have to rotate all their hand movements by 90 degrees to play a vertical flute, but the other body motions, changes to posture and breathing which subtly alter the sound of the instrument, have to follow suit as well.
I think to change this status quo, the most impactful thing to do would be to publish a free 3D printable vertical headjoint for whatever flute is most common in middle and high school bands/orchestras, and socialize the concept on platforms where high school and university music students are likely to see it. 3D printing is accessible enough, especially in academic settings, that it could capitalize on the "that's neat, I'd like to try it if I can get access to one" impression that a novel instrument tends to get from musicians. I think the collective feedback of a large group of intermediately skilled yet relatively adventurous musicians would show trends of how the vertical flute "really is".
Sadly, users would likely have to 3D print the entire head joint -- an adapter placed between the head joint and body of a existing flute would change the length of the instrument, and thus alter the pitch. However, 3D printing works ok for instrument parts which don't have to resonate too much -- it's been used for brass mouthpieces for awhile now, and can be comparable to other plastic mouthpieces if you finish the surface nicely.
That's a new and interesting fact to me! My mental model of why the keys on instruments change their pitch is that moving the open holes in the tube changes the length of the tube, which changes the wavelength of sound that's emitted. In retrospect, this formulation doesn't actually fit well with my model of "sound comes out end of tube" at all, but does fit well with "topmost open body hole".
Modeling the sound as coming from the first open hole does suggest another reason that formally trained flautists might dislike vertical flutes, though: changing whic... (read more)