WHY a belief in rebirth does not lead to the cessation of suffering?

nihilism ("non existence") and/or existentialism ("existence") must be avoided.

In the middle way, we accept all possible situation such as kamma causes rebirths, next lives, because we believe in causes and effects. But we do not believe in rebirth of self, because it is not the truth. It is just a rebirth of new effects, that may have the same look as in the past, but it is no self at all.

Q: But you're not saying anything different than I am Bonn. When we shed this shell and some other space is entombed in a meat-bag it's not the same, but it is the same. The space awareness was all one - except now it is another space covered by another meat-bag. You're saying the same thing I am yet you guys call my "continuity" to be existence. If MY awareness, that in my shell right now, were to be swapped with that of another shell I would still feel like the same observer but with different memories and perhaps varied senses.

A: I said "It is just a rebirth of new effects", but you said "the same". New is not the same. Your self view make you see the same. But new is new, new is not the old. A cause can not be itself's effect. Self causes not rebirth, self causes not self, kamma causes not self. Just new causes cause, new effects, rebirth. No self at all. Self is just an imagination of mind that is not smart enough to analysis the very fast ( the all time fastest stuff in the universe) of aggregates rebirth. So non practitioner always see "old aggregates" is "the same of new aggregates".

Q: I think you forget the cause and effect chain which results in karma. The end of the chain and its' beginning are the same.

A: No it is not the same kamma. In a second, we made over billion, uncountable, kamma. And in one person depend on uncountable kamma, and change every time. So it is not the same. No self at all. Don't try to keep self view. 

For the example, one's eyes born by wholesome kamma. Then these eyes are blinded by unwholsome kamma, after birth. So is wholesome kamma self? is unwholesome kamma is self? are regular eyes self? are blind eyes self? Or it is just new causes (that maybe arose in the past, too, and it still possible to be a cause of new effects) and new effects arising and vanishing?

This is the reason why paṭiccasamuppāda is very important in buddhism.