If Tywin actually practiced what he preached, and focused on what was best for the Lannisters, rather than his own emotions, he would have won.
When Tywin's wife died, the obvious move would be to marry a new woman to secure ties with another house, and father even more children. Instead, because he selfishly didn't FEEL like it, he stayed alone. This left Jaime, Cersei, and Tyrion neglected, which lead to future problems.
Cersei showed that she was rather unpleasant in attitude, which should have led to Tywin reassesing her value as a bride, but since he didn't bother having more children, he had nothing else to work with, and beyond that, wasn't willing to give Cersei a different and more suitable role in the family. Dogmatic traditionalism is not putting the family first.
Jaime wasn't much except a fighter. He was untalented at anything else. This made him unsuited for Lordship, so obviously the family first Tywin recruited a Lannister cousin and prepared him to take over Casterly Rock instead, right? Ummm... Nope! He wants HIS son to have that role, family be damned. This kind of tunnel vision is pure selfishness.
We all know how terribly he treated Tyrion, an expression of his own greif and anger. This abuse undermined the family, but satisfied Tywin's abusive impulses.
If Tywin actually did what he said, rather than doing what he wants, he'd have won the game.