It’s more of a consequence than a fundamental aspect. You need to see it in the following context, though:
When Buddhism started, the main religion in India was Brahmanism. It argued that when people died, their souls would seek a new body. Your actions in one life determines in what form, in what caste, you are reborn. The highest possible castes were that of clerics (brahmin) and nobles. In mainstream Brahmanism, you could not escape this cycle and effectively, the highest you could aspire to was to be reborn as a prince or a priest.
There were, parallel to this, ascetic sects that argued that by strenghtening your soul, to the great detriment of your body, it would become more independent and you could actually avoid being reborn, thus living a purely spiritual and eternal life.
Gautama was born a prince and he thought life sucked. Mainstream religion, however was telling him: “this is as good as it gets.” He tried ascetism, but, among other things there was the problem that there was no assurance that freeing yourself from the cycle of rebirth would free you from dukha.
Where freedom from suffering reaches freedom from rebirth is in the fact that in order to achieve the first, you must come to realise your true nature. That nature is that the ego, the soul, the atta is illusion. This view is called anatta. Once you fully realise the truth of anatta, freedom from reincarnation is logically inevitable as there is no soul to be reincarnated.
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