Think of it this way.

"I went back in time to save my family" in an infinite timelines story means that going back in time spawns in infinite number of worlds that didn't exist before, in which the family doesn't make it, and an infinite number in which they do. And not a single one of those families is the "real" family of the person who went back in time.

The fact that the author choose to focus on one perspective in which it seems like the time travel has made a difference, doesn't change the fact that it didn't make a difference, and the family they were trying to save is gone. The infinite copies weren't "saved" from anything, because there are infinite versions that weren't.

The only way to tell a meaningful story in that situation is to create situation where the actions of jumping back in time alter the future of the person jumping back in time. And that means you either suck up the paradoxes, or you write a clever story in which the paradoxes are neatly accounted for before they ever occur (or you write a closed loop story)