Yet by the time you reach the future everything leading up to it will have happened.

Whether the future is reached via speed, gravity or other ways all it means is that a place has been reached in a shorter time by the one travelling there than that experienced by those who experienced normal time passing.

There are a number of animals that can go into a period of suspended animation, some for decades, without ageing or experiencing the passage of time.  If you were put into such a dreamless state today then awoke in a hundred years time with your body in the same state as when it was suspended then you would have travelled to the future.  The same for someone travelling so that time for them slows down, nothing changes on Earth for those remaining there.