Wow… you just sparked something huge in my head (no pun intended).
Obviously (unless this is just some huge plot hole) the powers are not contained or produced in their bodies, otherwise Misa would not still have her power. So then the only other alternative is… its all in their heads. Not their physical brains, but their psychology. Basically, their powers come about because they think them up.
Im getting excited by how perfectly this fits as Im typing this. Think about it. Yuu says that all his life he’s wondered why he’s not someone else, and then suddenly he can literally become someone else. Nao, who wanted nothing but to be alone, out of the sight of the prying eyes of the scientists, can suddenly become invisible. Nao’s brother, who desires to create amazing music so he can make money for his poor family, can suddenly make perfect music without even having an instrument.
Now as to why all of their powers are imperfect. I actually am currently reading an LN called Utsuro no Hako to Zero no Maria (freaking amazing, btw), which allowed me to immediately see a reasoning for this: (I apologize in advance to all the people who won’t be able to read this part, but honestly you should just go read Utsuro no Hako to Zero no Maria right now anyway. It is that good.)
In Utsuro no Hako, there are boxes that have the power to grant literally any wish. However, the results of these wishes are almost always flawed, not because the box does not have the power to grant them completely, but because the wisher does not completely, 100% believe with all their heart and mind that their wish could possibly be granted by the box.
Thus, in Charlotte, their powers are incomplete because they do not completely believe that their “wish” (e.g. to be someone else, to be invisible) could possibly be granted. To even more from Utsuro no Hako, it is possible that the specific flaws of their powers are directly related to the imperfection of the wish. For example, lets hypothetically say Joujirou wanted to be able to teleport so he could escape from something. Thus, it makes perfect sense that he wouldnt be able to control where he stops, as his only objective was to get away.
There are also the several mentions of psychology in the on-screen words in the OP. Not only several mentions of parapsychology, but also some of those document file things say things like “phychological.” And remember: “It is a disease.” from the PV. Psychological disease?
And holy shit, just found something new: “the phenomena go away when skeptical ‘vibrations’ are present.” OF COURSE. (Same spoiler from above) Its like The Polar Express. The powers disappear when they grow up because they no longer believe at all.
Also, Charlotte is either almost certainly the name of the comet, or the name of the project of which all the documentation “were enclosed in envelopes.” Or both.
