I don't see why it is logically impossible to displace yourself in time any more so than it is to displace yourself geographically. My position and its effects on the world are fairly consistent whether I'm performing them at t1 or t2. If I can manage to transport myself to t2 from t1, it doesn't create a "paradox", I'm just now affecting things from that standpoint. Are you guys appealing to some sense of "fate" or "destiny"? I don't see why the future couldn't be altered. We know that the future can be altered. We do this all the time. We just usually can't see the effects of our actions until they transpire.
No, it's not about destiny. But suppose that you go from t2 back to t1, centuries earlier. So, at t2, you exist. At t1, you begin to pass on viruses that people at t1 have never encountered. One of your ancestors dies as a baby. Then you are never born. But how come you exist at t1? Two ways of looking at this that are consistent:
1. Parallel universes. You exist in your universe of origin at t1. The other one is different, and there is no counterpart of you in that one. The baby who died wasn't really your ancestor, but a counterpart of some sort in the parallel universe.
2. By going back to t1, you already destroyed the universe at any later time. So, there is no you at t2 (this is kind of a parallel universe scenario, but more destructive).
Are there other ways?
Yes, but either they do not affect the present, or they are very, very weird. And even in the weird ones, there is a problem that it's already happening, so the present gets destroyed from the past anyway.
I mean, suppose you go back to t1. Your very presence there changes the future, and somehow the changes propagate forward. Do you still get born? Even tiny differences can snowball. But in any case, with billions of years yet to come, other time travelers will go back and mess the time line so many times over that you get out of existence anyway. And I have yet to see a consistent model of that, which also is in line with our observations and does not result in some kind of global skepticism like the universe as we know it existed for a minute or a year or something like that.