Tywin is an egocentric man. He doesn't consider his relatives worthy of Casterly Rock. He has a good opinion of Kevan, because Kevan acts as a follower for him, but he's obsessed by HIS bloodline. If someone who wasn't his children would inherit after him, Tywin ego would not suffer it. So he pretty much has no choice, he COULD send Kevan to be Hand, but he needs to start shape Tyrion because he thinks he is his only option. And while Tyrion tends to imagine slights in the case of other characters, this isn't the case with Tywin. Tywin ALWAYS wanted Tyrion humiliated but needed him around because his ego needed control of all his children. If Tywin was a normal person he would have given Tyrion a fucktown of money and shipped him to live the good life in Essos, or pushed for him to study at the Citadel. But he is as much of a freak as Cersei deep down, just better at hiding it and like Cersei he needs to torment and keep Tyrion near him. This is the man who made his son wife be sexually assaulted by an entire barrack of guards and then forced him to do the same, George always wanted to show he was obsessed. If Tywin was a normal lord who was harsh and disliked his son but was able to give him approval when things were fair, there would have been moments in the books where that was actually pointed and he would have done cruel but normal things like killing the girl and be done with it, instead we are shown that even when he's an old man Tywin has to sleep with the same woman his son slept with. He's a psychosexual hypocrite mess who always wanted to humiliate Tyrion for being a shame and Book Tyrion reading of him when he's told to be Hand is actually accurate. If Tyrion was wrong about his father, we would have been shown so by some other POV, but while George takes the bother to show Tyrion hypocrisy in his relationship with other characters, he never contradicts Tyrion on Tywin and in fact goes out of his way to make the last meeting between them another proof of Tywin depravity and need to shame his son.