There's also another problem that Cyborg IM1's list didn't cover which is on my list of things: The power of psionics is immense - especially things like telekinesis and often work at great distances. All this requires work/energy and some method to manipulate it / produce it. I don't think "special radiations" within the brain would cover something like this. There'd have to be entire specialized organs - transmitters and receivers for this kind of energy. These would be would be pretty apparent upon autopsy and would certainly show up in genetic sequencing if it was inherited. There are no such a organs in the human brain and no humans have been found with such structures that nobody else has; so I think any explanation would have to explain that Earth humans don't have psionics. This makes things much easier because there's no need to "back justify" why we on Earth never found psionics.
But psionics do exist in humans, just not Earth humans. Psionics are closely associated with Droyne, the Ancients, and Grandfather, so I think the logical solution is that psionics are associated with that grouping.
IMTU, Psionics never existed naturally. It's wholly artificial, developed by Grandfather who sort of got tired of his physical limits in doing things. In the same way he discovered/created Jumpspace for Jump Drives, he discovered/created psionic space and found he could build "machines" in there which could interact with the "real" world in certain ways, subject to certain rules (while referred to as "machines" they are not machines of metal or plastic, and in fact are not made from matter as we know it at all).
At first, he limited access to this new toy to himself. It was great for him initially, freed from physical constraints, his unique intelligence allowed him to have dozens of telekinetic arms and remote viewing to attend to dozens of projects at the same time. Psionic tools were designed so they could painlessly and non-intrusively investigate things in the real world: the inner workings of still-living beings, subatomic structures, and so on. Even with such expanded ability, it still was not enough.
Grandfather tried to increase the number of his assistants to undertake tasks. First he tried having children, creating the super-Droyne species who’d later be known the Ancients. Then he or perhaps his children tried using human slaves for servants but finding them lacking in some way, he moved on. Determined to overcome this problem, he tried to genetically engineer servants and created the Vargr, though perhaps progress was lacking or seeing the limits of this project, he gave up on it as well, passing it onto his children. He finally settled on automations: robots, which finally were the loyal, competent, intelligent assistants he sought.
With each new group of assistants, Grandfather granted them access to psionics, crafting them the psionic tools they needed to be his assistants, for Grandfather’s projects and technology often required psionics to operate. A door with an opening mechanism on the inside that could only be manipulated by a psionic was very secure. His children figured out or were taught how psionics really worked and thus could independently grant psionics to the lesser servants as well.
Initially, when psionic powers were originally granted by Grandfather and his children to humans and Vargr, they were granted on a person-by-person basis and without the essential “copier” psionic machine (these are what allow psionics to be spread to someone else - you literally build tools onto the mind of someone else). These beings were used as overseers or trustees used to carry out sensitive duties or simply as group leaders who required psionic powers to access areas for their work. If one of these beings died or were rendered unable to continue their work, a new trustee would have to be selected and given a suite of psionic powers personally by the Ancient which made things more secure.
Once the Final War started, Grandfather’s children found themselves distracted by concerns of survival and war. However, most also wanted to continue their experiments and found they could no longer personally manage all of their projects and fight wars at the same time. They began granting psionic powers with the all-important “copier” machine to their helpers so they could do their work without oversight for years, decades, or even centuries. The copier machine allowed them to grant psionic powers to a group of beings as determined by the Ancient. Originally, the system meant that a group, be it a family or clan would all be on the access list to be able to use psionics so copied machines could be operated by the new psion. It was up to the psions in the family to decide who would actually be granted the powers. (Initially, Droyne populations only had limited members with psionics, but following the Ancient War, the species was struggling so much that Grandfather came by and granted them all psionics to help his species survive.)
Over the thousands of years since the Ancient War, interbreeding between those eligible and those not only have spread the potential to be recognized as eligible by psionic machines haphazardly, but far and wide as well. The end of the Ancient War marked the return to barbarity of all of these groups, and the knowledge of psionics, how they work, and how they’re transmitted have been lost to all but Grandfather.
What this means:
1. In this psionic space, which has slightly different rules to physical universe, but can still interact with it in certain ways, he was able to build psionic "machines." If you could see a psion's presence in this "psionic dimension" you'd see presence of the psion and from their heads, a cluster of various manipulators sprouting from them. Viewers to do this can actually be built at TL9 and mechanical manipulators capable of interacting with psionic machines around the same time. However, this can only be done if someone were to explain how these things work and described to you the device; nobody knows any of this and the true nature of psionics is a complete mystery, even to the Zhodani (who think it is genetic). Humans will eventually stumble onto the secret with greater knowledge of physics, around TL18 when exploring quantum subspaces.
2. Psionics are passed along by "paragenetic inheritance" - to prevent psionic machines from being accessed and manipulated by anyone, they have a genetic access key; only a person with the correct genetic sequence within a certain distance of the machine can manipulate it. At first this required an exact match of the entire genetic sequence. However, Grandfather's children were not as thorough and often needed groups of psionic servants, so took shortcuts, and made machines that only checked a limited number of genetic sequences for security. Through millennia of interbreeding, humans with the proper genetic sequences to use psionic devices are uncommon but not rare in many human groups.
3. Psionics only exist in Vargr, humans, and Droyne - all the races associated with the Ancients.
4. It requires a certain kind of thinking to manipulate psionic machines - they require training to use, and require certain thought sequences to function as a safety measure to prevent idle thoughts from making someone else's head explode. This alteration in thinking is why psionic potential starts high and lowers with age - as humans age their ability to alter their thinking to operate psionic machines diminishes.
5. "Testing" for psionic powers is bunk. What's actually happening is that the psionic "tester" is copying his or her machines to your brain - in essence, the very act of "testing" for psionics is what produces it. Of course, not everyone has the proper genetic sequences to use all the machines or any of them. When you learn a new psionic power, the teacher is copying their machine to you, and the rest is to see if you can learn to operate it or not or the machine recognizes you as a valid operator.
6. Not everyone can use the Copier machine - "bad" psionic teachers or testers are those without access to the Copier machine.
7. Because many versions of the same Machine exists; if you can't learn telekinesis from one psion you might be able to find another who can "teach" you it - it's not that your first teacher was bad, it was just that their version of the telekinesis machine didn't recognize you as a valid operator while someone else's machine does.
8. Because Solomani are the humans not selected to into the stars, they have the lowest rate of humans able to use psionics. However, due to the great genetic variety of Solomani, it's always possible through sheer dumb luck that someone might have the proper sequences. The Vilani are the second lowest population - all those Ancient Warbots and Vilani attempts to control them killed off most of their psions. Psionic incidence is greatest in human minor races and the Zhodani.
9. Some psionic powers are bad for you. While the powers engineered for humans by Grandfather were carefully designed for human brains, they were a very limited set - typically related to telekinesis and remote viewing/communication; powers necessary to use Grandfather's lab equipment. Later on, Grandfather's children created more psionic machines ("powers") that could be used by humans. Typically these were rush jobs by Grandfather's children and their adaptation is crude - the feedback from these powers often cause headaches, lowered mental functioning, fatigue, insanity, permanent brain damage, or even death after a period of use (the length of the time and how often you can use them varies). However, these powers are more interesting (from a player point of view) including offensive powers like firestarting, teleportation, and so on.
10. Psionic helmets disrupt the "MRI hyperimaging" psionic machine's ability to read data. It was never intended to be used against active resistance so it's easy to disrupt. Putting a helmet on a psion disrupts the ability of the machines to read the brain's commands (and to communicate data back) ... all of these signals occur at very low, safe-for-the-human-brain power levels after all. Psionic powers for that matter, won't work near particularly strong EM fields for the same reason.