By cessation, I take it you mean cessation of consciousness?
Cessation of consciousness was not the goal in the Atthakavagga. Cessation of sensory perception was. Suffering was caused by contact and clinging to the world. Cessation of sensory perception was the cure.
Cessation of consciousness emerges as the goal in the Parayanavagga. Suffering was caused by a notion of self. Cessation of consciousness was the cure. This seems to have been a major schism in early Buddhism. Atthakavaggan Buddhism was agnostic toward or independent of metaphysical views.
Snp 4.3
There is not in the world such a purified person
who continues in these views about existential states,
Parayanavaggan Buddhism definitely had metaphysical views that pushed it toward the goal of cessation of consciousness.
Ajita
So wisdom it is and mindfulness!
Now, sir, I ask you, tell me this:
the namer-mind, the bodily formâ
where does it cease to be?Buddha
That question asked by you
I tell about it now,
the namer-mind and bodily form
where they cease to be:
by cessation of the consciousness,
they wholly cease to be.
This move to an adherence of a metaphysical views and the goal of cessation was mentioned and NOT endorsed by the Buddha in the Atthakavagga. After stating that the goal is cessation of perception, the sutta continues:
Snp 4.11.
Question
Whatever weâve asked of you, to us youâve explained,
another query weâd ask, please speak upon this,
those reckoned as wise here, do they say that
âpurity of soul is just for this (life)â
or do some of them state thereâs another beyond?Buddha
Here some reckoned as wise do certainly say:
âPurity of soul is just for this lifeâ;
but others who claim to be clever aver
that there is an occasion
for what has nothing leftover.And Knowing that these are dependent on views,
having Known their dependence, the investigative Sage
since Liberated Knows, so no longer disputes,
the wise one goes not from being to being.
Clearly, views carried the day in Theravada Buddhism, but that was not the case is the proto-Buddhism of the Atthakavagga.