I did a time-travel plot in my Rogue Trader game, recently. It started as 'it would be funny for the PCs to encounter two versions of Ahriman fighting each other' and spun off from there.

As a result, I had to do a lot of thinking about how I wanted the time travel to work, and what would change if they actually succeeded. I ended up going with divergent timelines. Their efforts to change the past spun off a new version of the universe where they had already changed the past, and one of the knock-on effects broke the time machine they used to do it so they couldn't change things again.

That's for an RPG though, where I want the decisions the players make to actually matter. If I was just writing, I think I'd stick with stable time loops. You can't meaningfully change things that already happened, because they already happened and you were already there. You might end up recontextualising some events for yourself, but ultimately you've already done whatever it is you're going to do.