I like Prallax's answer, and as a humble geophysicist, all I can contribute is the way I visualize how causality in closed timelike curves works. Buckle in, cause it takes a whole story for me to wrap my mind around it.
Firstly, we're talking Harry Potter book 3 style time travel. As in first, Harry (let's call him Harry1) sees a dementor, and then sees a "mysterious figure" who casts a spell to get rid of the dementor, thus saving Harry1's life. Later, Harry1 uses a timeturner to go back to that moment and becomes that "mysterious figure" (Harry2) who saves Harry1's life. So basically the grandfather paradox is avoided because he saves his own life.
What I believe (but I think this is still unproven) is that the grandfather paradox does act as a fate-like force, essentially preventing any actions within the closed timelike curve that would lead to the death of your own grandfather. You must follow a path of choices that allow you to enter the closed timelike curve or use the timeturner in the first place.
In Harry Potter, this means that we can predict that Harry will go back in time to save his own life, but we can't predict how he will do that. He may use a patronus spell. He may grab Harry1 and disapparate. He may ask Hermione to cast a patronus spell instead. He may cast a patronus spell 3 seconds before the dementor gets to Harry1. He may cast it 3.1 seconds before. Or 3.2 seconds before, or 3.15, or 3.125 etc. He may even sacrifice himself to save Harry1. He can do literally anything (within the laws of his universe) so long as it allows Harry1 to survive long enough to use the timeturner. Ie, there are infinite possibilities for what will happen, but it is a limited infinite. Just like there are infinite numbers on a number line, and infinite positive odd numbers, but the latter is limited infinite, as they exclude even and negative numbers. We have not narrowed Harry's options down to a single possibility, or even a finite set of possibilities, but we have limited the set of possibilities.
We know Harry2 will not go back and stab Harry1 to death. We know Harry2 will not decide to take a nap and forget to stop the dementor. All possibilities that prevent Harry1 from using the timeturner are cancelled. They are physically impossible, and any universe or timeline in which that happens will effectively cease to exist, and we experience them as never having happened in the first place.
From Harry2's perspective, that might feel like "fate" or "God" nudging him in certain directions. Harry2 might really want that nap, and as soon as he falls asleep have a "prophetic" dream telling him to wake the f up and go stop that dementor. He may try to use his free will to ignore that dream, but again him not stopping the dementor is physically impossible. So he may then experience some wierd stuff that sure feels like a "miracle" or "magic" that forces him over to the dementor. That "miracle" will have been a technically possible though normally highly improbable series of coincidences, that was "forced" to occur in order to satisfy the timelike curve. (Or rather, because it is probabilistically possible and does not defy the grandfather paradox; and any other storylines which do not meet those 2 criteria cannot exist, this is the storyline that occurs). And had he just avoided the nap in the first place, there isn't even a need for "miraculous" forcings. That's a free will discussion for another time.
I believe Harry1 will also experience these forcings. He does not have the choice to opt out of using the timeturner, because the loop has effectively already happened. He is already in it. Even the "first time through," he saw the "mysterious figure". From his perspective, time travel has already happened, and he can't get out of that loop. He has to go back and save himself because it already happened.
Entering into a closed timelike curve places limits on what possibilities can unfold in your future, specifically during the loop (the time between Harry2 appearing in that past and when Harry1 uses the timeturner). Just those chapters: 16-22. But as for what happens in chapter 23 or book 4, the possibilities are just as limited as they would normally be. Harry would have only experienced "fate" during the loop. Not before or after. That timeloop exists as a defined event in the past now, just as surely as any other life choice Harry made in book 2 does. It affects the possibilities of the future, just as Harry's deciding to go to Hogwarts affected his life opportunities, but no more or less so than that. If you are starting at book 1, trying to predict the whole series, the fact that a timeloop happens in book 3 is just as predictable as any of the other plot points. It's only within the timeloop that predictions are limited. And since it is a complete loop and doesn't "change the past", it does not add any extra unpredictability or uncertainty to the series as a whole.
In conclusion, within a closed timelike curve we can predict parameters or limits on what will happen, but the specifics have infinite possibilities within those parameters. It would be like being able to know for sure that you are going to live to 92, but having no clue how you die. Or how you live your life for that matter, other than knowing you will be prevented by "fate" from entering a deadly scenario for the next 60 years. It's not even that big a deal to know some parameters of the future, as we already have that. I know for sure I won't live past 200, for example. It's just a different set of parameters. Outside the curve, the possibilities are the same as usual.
Disclaimer: I'm not trying to push any religion into this. The use of "God", "fate" and "miracles" is merely a placeholder, as I don't have the vocabulary to describe these things. You do you in terms of beliefs. Also, I really only have cursory knowledge of timelike curves, haven't been able to work through the proofs or anything myself, so take this with a grain of salt.