Unpopular Opinion

Hear me out. There's a lot of signs that I see that Tywin is actually trying to help Tyrion. The key moment is when Tywin yells Tyrion that he is worthless and doesn't anywhere near the family or Kings Landing and that he should disappear and do his own thing. Afterwards Tywin forces Tyrion marry Sansa that would have not only secured Lannister and Baratheon support to secure the north, it would have made Tyrion a lord, and a lord to one of the most honest and noble houses in the entire mythos.

Tywin never made any claim to another person that his son disgusted him Tye only time he would same something of that sort was in front of Tywin himself. He would always defend him as a Lannister no matter what. Tywin was even well aware of Shay and let it go until the king died. Tywin was calculating and wise. He would know how to talk to his child to guarantee they seek their own path. Another show of this is at Tyrion's trial. Jaime sacrifices what he wants for his brother and Tywin accepts it without hesitation.

My theory is Tywin was never meant to hate Tyrion, but knew that Tyrion could be doing greater things than the politics of his house, and wanted Tyrion to be free of the house because that would be the only way. Tywin even tells the story that he was going to throw Tyrion into the ocean, but he didn't. He let him live. He never explains why he let Tyrion live. He just did. Was it love for your child? I think so. I don't believe Tywin wanted the same life for Tyrion, but thanks to Cersei, Tyrion was just pushed too far