Being smart doesn't shield you from irrational hatred though. Having a dwarf as a child is super shameful, especially in a noble family. Everybody hates and mock dwarves. Add to that the fact that the beloved spouse died in childbirth adds injury to the insult. Tywin Lannister said it himself in a fit of rage when Tyrion confronted him.
I'm more interested in how you managed to clue that strange and interesting theory that the Lannister trio aren't Tywin's. I can't see anything that would hint at such in the whole 7 seasons.
Jaime and Cersei feel bound by some bond they can't explain to have incestuous relations with each other.
Cersei in particular is prone to fits of rage and constantly talks about burning things to the ground, and eventually does.
Jaime views his sister with the same amount of growing distrust as he did the mad king.
Tywin says he wanted to carry his son into the sea on the day he was born.
Tywin tells Tyrion "You're not my son." Now if he meant this as another way to say "I hate you.", why not just say that? He may well be telling the truth.
Tywin was hand of the king to Aerys Targaryen and was forced to obey his every whim even as he slipped further into madness. It would not be a stretch to suggest that this arrangement included bedding Joanna Lannister, Tywin's wife and cousin.
Tywin will not make Tyrion the heir to Casterly Rock, opting instead to end his own line rather than let him do this. Again, simply being a dwarf doesn't explain why he thinks this way except that he's not really a Lannister.
Tyrion has a mysterious affinity for dragons. Everyone else who comes near dragons is met with hostility, but Tyrion immediately warms to them.
Robert wins the rebellion and kills the Targaryens but is uncomfortable even around members of his own family.
Tywin is overcompensating for his own inability to produce legitimate offspring by constantly berating his "children".
Cersei and Jaime are both innocent of the crimes perpetrated by Littlefinger and Lysa at the series' impetus. The Starks and the "Lannisters" are thus part of the same game, two sides of the same coin.
Dany feels the need to almost instantly trust Tyrion despite the fact that they barely have anything in common.
Tywin hates both the Baratheons and the Starks but is more than happy to destroy Aerys' entire family with their help.
It's a tinfoil theory that will never happen on the show because they have no time, but had it actually been true, there would be no evidence disproving it either.
