To reach any conclusion about time travel you would first need to understand time. It is easy to say that time is constant, the minutes, hours, days, years, tick away, at a regular pace. But that is not true. You know yourself how much longer a year was when you were young and how much quicker it passes when you are older. Your younger years were possibly marked on the wall as you grew taller, but it took forever for time to pass.
Time is perhaps a necessary concept for order, however it doesn't necessarily exist. And if it doesn't exist, or it doesn't neatly mark out our passage through life in an ordered, measurable way, then what would be time travel. You would go back to somewhere before, when there is no before, except in your memories which are like dreams, hard to make solid, distorted, and not real.
How about an ever changing now? No past, no future, it only seems like there is when actually there is only a huge now! Time makes sense because we physically grow old, if we didn't age it would make no sense at all to have time. You can't travel to a place that doesn't exist, so you cannot travel back to a past that never was. It is perhaps possible to cut through the now, but cutting through now is simply making connections and there are a huge number of those and they might influence the future, but not if they were there all along. Not if there is no future only a changing now, which seems like a future, but isn't.
Time travel does not exist because time does not exist, you can't travel on a concept! And yes, to a certain extent destiny and fate are real enough, but you don't properly know your destiny, you might recognise it when it arrives, let's hope so!