I feel like the portrayal really depends on the era and social class the saga focuses on. A lot of the classics, think something like 'The Forsyte Saga', give us a man who is primarily a steward—of the family name, its wealth, and its reputation. His emotional world is often secondary to his duty. He might be a remote figure to his children, a negotiator in business, and his marriage is frequently a transaction. The drama usually comes when personal desire clashes with that rigid role.
That said, I'm more drawn to the ones that subvert this. Not every patriarch is a cold monument. In some recent sagas, you see the strain of that position. He might be privately insecure, aware the family fortune is built on shaky ground, or struggling to connect with a son who wants a different life. The 'man of the house' facade starts to crack, and that's where the real story is for me. The performance of control is often more interesting than outright, unchallenged authority.