I am going to address the second option on your list, Many Worlds, in more detail. In my opinion, it is the only physically-sound explanation for how time travel could exist while no effects from it would be visible. That said, Many Worlds is itself a bit of physics-woo, so make of it what you will.
The basic concept of Many Worlds is that for every possible outcome of an event, the universe splits, and both (or however many outcomes are possible) occur.
It is closely related to the concept of Quantum Immortality, which comes out of the application of Many Worlds to Schrodinger's Cat. In that thought experiment, a cat is left in an undermined state inside a box, either alive or dead. The primary issue is that, if the cat is in an undetermined state, what does the cat perceive before the box is opened? Many Worlds suggests the cat experiences both continued life and death; Quantum Immortality is the suggestion the cat would merely experience continued life, thus becoming immortal.
This is related to time travel in that it suggests every possible event, no matter how incredibly unlikely, occurs. Considering phenomena such as zero point energy, where probability is the main factor preventing absurd random occurrences (and entropy, the addressing of which is beyond this answer), the word "possible" can effectively be removed from that statement: every event occurs.
Now, imagine some time traveler returns from 2222 to now. Let's presume the time machine comes with them, for the sake of discussion. What is the probability that a time machine and a time traveler appears in your front yard in the next hour? It's insanely - nearly impossibly - unlikely. But following certain conditions and making certain assumptions, it is possible for that to happen. Meaning in some universe, it must happen.
Now, there is an issue with this, and it is where the cat returns. From your point of view, the probability of that occurrence is so vanishingly small it is easy to ignore. People and time machines do not just materialize out of thin air. But the probability from the time traveler's point of view is substantially higher. If the trip does not fail and he is not killed in the process, his only option is to appear in your front yard.
It's worth pointing out this would suggest probability can be "localized" to a given person. Again, that's well beyond the point I am making.
Because of the difference in the probability of outcomes between you and the time traveler, we end up in a case where the time traveler is effectively routed to a specific branch of time where he and his machine popped into being. You and I never experience this because it is so insanely unlikely that just in a pure probabilistic sense, we end up in some other branch of time. However, at least some number of you in other branches of time do see the time traveler. There's just many more of you that do not experience a time traveler appearing in their yard.
It is also worth pointing out this would mean temporal paradoxes are actually impossible to form, since upon going back in time, the time traveler would be entirely decoupled from their own origins. They effectively just spontaneously came into existence remembering "their" future.
So to summarize all that physics-woo, the Many Worlds interpretation applied to time travel would suggest it is merely very unlikely we end up in the same universe as any given time traveler. It's not impossible, but so insignificantly unlikely we can go about our lives with it never actually happening.