In some respects, the idea that somehow a new "story" is written (By whom? the Guardian of Forever? The mind of the Creator of the Universe?) each time The Time Traveler tries to save his fiancée in which she nonetheless perishes is profound and I don't think any other version of Wells' book has this idea.
However, could not Alexander have made his machine because she was simply badly frightened, etc.?
Not necessarily, if you figure out a clever way of faking her death. And most likely to move her to the future to avoid any unwanted past interaction. That way your past self would act in similar/same way, which would be perfect outcome.
Similar in:
Back to the future, where Marty would not write any letter if he saw Doc simply surviving/avoiding death. But he saw it, so he acted like Doc was dead.
This ... was not a great movie, but if we take it's twist at face value, then no, there was no way to prevent Emma's death.
The premise of the movie, is Alexander's quest to find the answer to question, which I am possibly paraphrasing: "Why can't you change the past?"
And the answer he finds at the end, again a paraphrase, is: "Since you invented time machine, to change the past. The past can't change, otherwise you would not have invented the machine to begin with"
It is not an original time travel paradox solution but, it is the one the movie goes with and for what's it worth, this aspect, is portrayed well. As is, it implies, that should you change the past, the past will correct itself in a way that prevents the paradox.
While yes, Alexander could have invented time machine for different reason, the logic of the paradox would remain and changes should therefore not remain.
