The Buddha never denied the existence of a person that is born and dies.
On a conventional level we can say person, as a way of reference
However in reality there is no "person" or "me" or "you"
It's an illusion created by clinging
Sure a personality can remain, if by that we mean habits and thought patterns, but they are not-self either
So if we do not cling to "personality" or the body etc, there is emptiness and non-death
There is no more "I am this" that dies, there is just the way things are in the moment
“So, brahmin, when there is the element of endeavoring, endeavoring beings are clearly discerned; of such beings, this is the self-doer, this, the other-doer. I have not, brahmin, seen or heard such a doctrine, such a view as yours. How, indeed, could one — moving forward by himself, moving back by himself — say ‘There is no self-doer, there is no other-doer’?”
When there is pure awareness then there are no "beings"
However when there is ignorance, then beings are born who endeavour/intend
"Then, Bāhiya, you should train yourself thus: In reference to the seen, there will be only the seen. In reference to the heard, only the heard. In reference to the sensed, only the sensed. In reference to the cognized, only the cognized. That is how you should train yourself.
When for you there will be only the seen in reference to the seen, only the heard in reference to the heard, only the sensed in reference to the sensed, only the cognized in reference to the cognized, then, Bāhiya, there is no you in connection with that. When there is no you in connection with that, there is no you there. When there is no you there, you are neither here nor yonder nor between the two. This, just this, is the end of stress."
http://www.accesstoinsight.org/tipitaka ... .than.html
"there is no you in connection with that"
So when we are fully aware of the present moment then we are no longer deluded by feelings, we dont chase after pleasant feelings or avert from painful feelings. We then do not cling to that which dies and do not give rise (or birth) to "I am", which is associated with that which dies.
So we are not connected with that which dies. If there is no connection with death, there is non-death, deathlessness or non-experience of death
There is also no ageing, sickness or dukkha in general
However if we are connected to that which dies, through clinging, then there is ageing, sickness, death and dukkha in general
"Now this, monks, is the Noble Truth of dukkha: Birth is dukkha, aging is dukkha, death is dukkha; sorrow, lamentation, pain, grief, & despair are dukkha; association with the unbeloved is dukkha; separation from the loved is dukkha; not getting what is wanted is dukkha. In short, the five aggregates subject to clinging, are dukkha."
Or to put it another way
In short, the five aggregates subject to clinging, are dukkha. This leads to Birth which is dukkha, aging which is dukkha, death which is dukkha; sorrow, lamentation, pain, grief, & despair which are dukkha; association with the unbeloved which is dukkha; separation from the loved which is dukkha; not getting what is wanted, is dukkha.
Instead we just experience things as they are, without reference to "me"
"When there is no you in connection with that, there is no you there. When there is no you there, you are neither here nor yonder nor between the two.
That Is there is no "I am this body that dies", there is no self and so no death or dukkha. There is no self anywhere that can die.
This, just this, is the end of stress.
So when we are truly mindful, there is no dukkha any more and so no death