I think i figured out how it works both Basilisk Stare in eye works, Harry Potter Ghosts and bit of soul parts etc. I am going to give you 4 Reasons/Clues first.
Clue/Reason 1
1 The basilisk stare turns you to stone when you catches its eye in a reflection, glass/water etc... the petrified student does not need to do the things you do everyday I am talking of things like eating, going to toilet sleeping, Socialising. But people does not 'forget' you exist or anything or existed but you are in essence like dead to the world.
Clue/Reason 2
It kills when its eyes meets its target now why is that? one guy talked about a 'beam' and that its always on but I am quite sure thats not how it work but i believe the guy was not completly wrong I think he is in the right direction is a sense.
Clue/Reason 3
As everyone probably knows/heard of this And yes I am taking a quote not from HP here: The eyes are called 'the window of the soul' -William Shakespeare.
So the 'soul' is basically The 'core' of who you are, and it is said that the soul can be seen through the eyes.
Clue/Reason 4 and last.
If we look at the ghosts of Hogwarts specifically ghost Nick he got turned to stone it was no different really compared to the 'living', the basilisk stare in a sense even sort of harms the soul it simply must be like that, there is no other alternative. HP quote: What could possibly even harm the dead?
With these 4 Clues etc I will now form a hypothesis consistent with several parts I believe is true to each part of how the gaze works etc.
Hypothesis part 1
So what if its true gaze is actually not severing the soul or making someone want to suicide etc? But does instead freeze/petrify the soul? A frozen soul in a body would it not mean the body would shut down in a sense? Could it not act as if it was severed? If so then it would to the body be as if the soul no longer exists in it since it can't sense the soul.
Hypothesis part 2
Now how long would/could the soul or a ghost stay frozen? Well a ghost in my theory is a spiritual construct that is sort of a clone of who the real person was in life but possibly with a link to the soul in afterlife. So probably as long as the magic is around id say/believe.
Hypothesis part 3
In a body the Soul probably would not stay frozen/petrified for long in a body I believe because it would be ejected eventually and the magic would break, the body is said to be merely a container after all and the basilisk stare hits the eye first thus magic should be centered there if you will. Thus the magic is most likely circling or so in the body and once the soul gets ejected the magic stays in the body not following the soul. And when it is about to hit the soul through a 2 reflection like Mirror/water then to the eye, the effect is simply cast upon the body instead Exactly why is something of a 'mystery' but a Hypothesis here is coming
Hypothesis part 4
what if to the magic of the basilisk gaze, 'the eye' of a person/being/ghost is also magically ruled as a reflection to the soul? And if we take that into consideration since the mirror/water reflection was 1 what if the gaze is simply not reaching the soul and instead the magic gaze affects only and on the body instead of the soul? This would make the body not dead, but like frozen in a sense the soul is still linked-up/tied to the body and therefore the person does not die.