Since Lovecraft is problematic,
and since insanity is hard work to design sensitively and accurately,
a different approach seems useful.
Personally, I tend to view Aberrants as Neutral Evil, as the missing species between Devil and Demon, seeking to disintegrate reality itself, namely the Material Plane, and doing so by means of evil wishes − and reality warping.
Aberrations are both the consequences of disintegrating reality, and the vehicles of temptation granting wish-like capabilities, at a price.