Hi Ernic,
Thought I'd chime in, with some observations of my own, based on singing with these instruments:
there might have been two general types of hurdy gurdies: one with a rather smal body for religious and more serious music (especially for accompanying singing?) and one with a large body for folk/dance music (functioning as a parallel of the bagpipe). Both in several shapes. Is this just an impression, or might it be true or is it false?
My thoughts regarding size (and of course this is related to the shapes/layouts): The size of the instrument is directly linked to its resonance, tone, and affects the placement (ie. distance) of the wheel vis-a-vis the bridge, affecting timbre and tone to an extent. This then makes a difference as to how the voice interacts with the instrument. Taking this into account, adding to it if we presume that Odo's reference to an "organistrum" in his attributed "quomodo org. construatur" is indeed one of the larger instruments, closely associated with the ecclesiastical world, and then adding as well that empirically speaking, the smaller sinfonias are almost always associated with the courtly milieu, it might seem that the following is somewhat the case:
up to about 1300/1350:
Organistrum (either 1- or 2-person): predominantly depicted in religious scenes/perhaps in religious use
sinfonia: almost unfailingly depicted in a courtly setting (Cantigas is courtly, though dedicated to Virgin Mary)
from about 1350/1400 onwards:
keyboard on top or closer to wheel, 1 person, more "bosch-like" instrument, depicted in both sacred and secular settings
very few organistrum-type layouts seen past 1400s.
Now, as to singing with the instrument, the character of singing is quite different, of course: a larger instrument, with its different resonance (usually a bit mellower, a bit darker, less nasal) lends itself quite well to voices (especially the longer string lengths of organistrum, to male voices--given the register) and to accompanying chant and medieval song. Your Aachen, and my Burgos-Leon instruments, are examples of such instruments. They conform to that organistrum-shaped body. The tessitura or primary range of many of these instruments--which can of course be changed by restringing--seems to lend itself well to live around G. With regards to resonance, there are of course certain soundbox sizes and shapes of instrument that lend themselves better to certain pitches.
The timbre, color of the tone, etc. of smaller sinfonias usually lends itself better (in my experience) to minnesang, trouvere chansons and troubadour cansos, sung virelais (ie. the cantigas are an example) and other music with courtly ties. My hypothesis here is that it is based on the manner of textual deliver as well, since a delivery closer to speech, such as chansons and minnesang, would tend to compete more with an instrument--so it would make sense for that instrument to have a more forward presence.
These are just a few thoughts, based on my experience with the instruments and singing medieval music. Becuase what enchants us on this list is the instrument, it might sometimes slip the mind that the primary instrument in the middle ages was the voice, and most other instruments work with it or emulating it, the few exceptions of known instrumental music notwithstanding.
Tanks again for your interesting thoughts, Ernic! (PS: I'll be around NL in May with Sequentia on the Oudemuziek circuit, give a shout if you will be near Maastricht, Enschede, Rotterdam, A'dam, or Utrecht!)
All best,
Vlad