So the point is is that, in the show, once he's sprung from prison, he's elated that he's escaping. He had given up and now he has hope again. He even has a heartfelt goodbye with Jaime. The sudden change to anger was a little unnatural. It's like he had to stop and remember, oh hey, I should take care of some business before I leave. The Tysha story explains why he's feeling angry at that moment. (Yes, he has plenty reason to feel angry at any time in his life, but in the show nothing provoked it right then.) After his talk with Jaime in the books, he's angry, he's hurt; he wanted vindication and he doesn't care about anything else.
The part about Tyrion valuing family is maybe only from book Tyrion. It's hard to keep the two separate in memory.
In general I agree with you that "hey this adaptation did this differently and therefore it must be worse" is poor logic. It's hard to judge in this situation because I can't see the show from a position of not having read the books. However, I do think here that his motivation was not as compelling.