To prevent spreading of special Ascension traits from being overpowered, maybe move the majority of the stat bonuses off the actual trait and have it as an Empire modifier. So base Cyborg could give maybe a +5% Habitability while an empire with "Flesh is Weak" gets a stat that grants the remainder of the buff to all pops with the Cyborg trait. Same thing for Psionics as well.
The explanation being that while other empires can replicate the trait, they lack the understanding and infrastructure to really support it. Cybernetic empires have a better ability to customize and adapt cybernetics to meet the user's requirements while psionic empires have a strong grasp of what being psionic actually means and how to exploit it. Essentially, other empires could get still get the traits, but they'd be cheap knock-offs compared to empires actually pursuing that route.
[EDIT]: Hm, could you link that with Espionage, maybe? Unlock the ability to spread the trait via espionage stealing the basic knowledge from another empire, and maybe increasing the bonus slightly with increasingly difficult operations. Never to the point where it can match an empire with the Ascension perk, but it's something.