Hi again,
The reason I asked about the consciousness of an enlightened being was since you said
One way to understand the difference between true cessation and mere cessation: when merely ceasing one thing necessarily gives rise to another. That is opposed to truly ceasing where if something truly ceases - that true cessation - cannot be said to act as a condition for the arising of some other.
and usually the consciousness of an enlightened being is said to disappear so thoroughly that no being in existence can locate it. In that sense something has truly ceased as you define it without giving rise to something else but was not substantial for the being themselves. Perhaps you interpret that it was substantial for all the non-arahants hanging around 
so it only ceased truly for them but even the arahants would not be able to find it, I suppose.
On,
Continued existence of what? Ignorance and craving.
Just wanted to say that I’m not sure if the term Continued existence (bhava) is stretchy enough to accommodate such a statement. From what I know, bhava refers to existence in a life and enlightened beings do have bhava while they are alive as mentioned in SN 22.76.
As far as there are abodes of sentient beings, even up until the pinnacle of existence, the perfected ones are the foremost and the best.”
Yāvatā, bhikkhave, sattāvāsā, yāvatā bhavaggaṁ, ete aggā, ete seṭṭhā lokasmiṁ yadidaṁ arahanto”ti.
So it is directly tied to rebirth, instead of referring to the Continued existence of a view of a true self. But bhava does cease with death for the enlightened ones since it arises based on ignorance and craving. For more details, you can check out this thread too.
Maybe SN 12.61