The Blackfish - somewhere you have to draw the line and make the stand, regardless of how futile it is. In the end you pick something you'll both fight for, and more importantly, die for. For the Blackfish it was his home.
I don't buy that. It's totally against his character. He was
not the Lord of Riverrun, just a temporary warden. His nephew is. Which was made very clear to him by the garrison. Also, he's been away from "home" for most of his life and served House Arryn since Robert's Rebellion.
Dying senselessly is not "fighting for his home". Especially if there is a way to win it back someday, with the help of the North. Just wait until Jamie takes his army back to King's Landing, and take back the Riverlands, and destroy the Freys in the process...also something the "real" (=book) Brynden Tully would have had in mind: there's a
lot of unfinished business with the Freys.
At least in the books he did the smart thing.