I decided to make my own go at homebrewing some 5e Wild Psionics rules, we are using Korranberg, but I am allowing other psionic abilities from other books to be used if the player wants. Nothing is random, the player chooses the abilities they can have. Let me know how you feel about it in a 5e context.
Here is a copy and paste of the homebrew item. I have it in pdf if anyone requests it, I can send them it in email.
(We are using Korranberg’s Psionics for my current campaign, but I would allow abilities from The Will and the Way, or The Complete Psionics handbook or another material if the player runs it by me.)
Whenever a character that is not trained in psionics survives a targeted psionic attack or ability, that character gains psi points equal to the amount of psi points used against the character to a maximum of 5 psi points.
If the targeted Psi attack or ability on the character costs more than 5 psi points, the character must roll a mental ability save check -the amount of psi points above 5. Example: a character survives a 7 PSP attack, the character can absorb up to 5 PSPs but must make a Mental saving throw at -2. Some Dark Sun items may temporarily increase the number of PSPs you are able to have. If this is the case with your character, madness does not set in unless it goes over 5 + the temporary PSPs.
In the event the non-Psion character fails the check, the character will develop a Madness at the DMs discretion. The DMG gives a chart for this if you wanted to use it, also the Curse of Strahd has a similar chart.
Your Mental AC is your highest mind stat from Wisdom, Intelligence, or Charisma. This is your Primary Psionic Stat that you will use for all checks using Psionics. When attacking with a Psionic ability, you use your highest mental stat as your attack modifier. If the psionic ability you are using is your wild talent or wild augment, you may use proficiency. If you are using a psionic ability that is
unknown to you through the use of an item or other odd circumstance, you may not use your proficiency.
If you are using a Psionic ability that is not an attack or does not require the target to roll an ability save, you must roll below your highest mental stat for the effect to manifest. You may subtract your modifier from Your primary mental stat, and subtract your proficiency, if you are proficient in the ability.
At any time during a short or long rest, an untrained character that has psi points may expend psi points and buy 1 psionic augment or 1 psionic talent (psionic cantrip). The augment or talent will become that characters wild psionic ability that your character has awakened. No other augment or talent can then be learned unless the character multiclasses into a psionic class, or has another
ability or feat that grants you an augment. So an untrained character may have one psionic augment and one psionic talent, unless another ability or feat gives them more than one.
Once you have awakened your psionic ability, After a short or long rest, you will always generate psi points up to a maximum of 2, you can never generate more than 2 points on your own, unless you are trained in a psionic class, or unless another ability or feat grants you psi points.
Once you have learned at least one psionic ability, to gain more than 2 psi points the character would need to absorb psi points from being psionicly attacked or having a psionic ability used upon them up to a maximum of 5 psi points and survive.
So in the case in which an untrained character bought a 3rd level psionic wild augment, which normally costs 5 psi points to use, they would only be able to use it if they absorb psi points (or had an item or ability that increases temporary PSPs), but if the same character bought only a 1st level psionic ability the character could use the 2 psi points they generate naturally to cast the psionic ability, without needing outside help.
Trained psionics users do not absorb psi points like non-psionic users, this is due to the training of their mind, they generate enough psionic power on their own that they don’t need to.
In the Dark Sun campaign setting trained Psions are given 2 extra PSPs at first level, and may learn 1 extra talent, and 1 extra Augment. If the character has multiclassed from a non-Psionic class and has already learned 1 wild talent, or 1 wild augment, the character does not receive the extra psionic abilities."