The person didn't die, because it wasn't killed in a traditional sense. This point of the story is highly philosophical in nature. You may have noticed the Night Watch cycle getting more and more philosophical throughout the books.
What is the difference between mortals and Others? What makes an Other an Other? Can a mortal become an Ohter? What is the Twilight?
To understand this, you have to remember what the cause of 'death' was.
The entity representing the Twilight
hasn't killed the person, but
the Other in him.
It wasn't a physical event, but a 'magical' one. Every Other which is 'human enough' could have survived. This wasn't stated, but can be concluded from the statements of the other characters in the end scene. From this scene, we could also strongly assume how the 'death' was achieved.
Anton's 'magical temperature' was raised so high, he can never hope to become an Other again. So the Twilight entity seems to have used kind of an inverted Fuaran to 'kill' him.
Following this thought, we end up at a point to admit that
if Anton wasn't killed and just stripped of all his powers as an Other,
every Other could have survived the event itself. 'Surviving' has a deeper meaning here than 'come out of an event alive'. The person 'survived', because
he has preserved a human mind and can still accept to live a life without supernatural powers.
This conclusion gives a lot meaning to all the passages about using magic for mundane tasks, found throughout all six books.
So many times Anton explains something in the style of 'of course I could have used magic for that, but I decided to do it by hand'.