I think I have an ultimate recipe for solving time travel paradoxes in fiction while retaining most of the perks:
what if there are many other universes, and the only difference they have with this one and each other is the initial moment in time?
The lack of actual time travel automatically eliminates all the issues of time travel like loss of information or consistency issues. And it removes the conceptual issue of time travel of it needing a backlog of the universe to get an earlier state, which is a bit arrogant to think the universe would have for our time travelling needs. So no human constructs like timelines.
Instead it's a series of independent universes that just happen to align through the identity of initial conditions. Like what happened in Futurama but in parallel and with offsets.

Not that I would like to write a novel, I just find it odd that, to my knowledge, this concept isn't already floating around.
Maybe I'm ignoring some great work of fiction, that I wouldn't know :-).