I can see how one might about conveniently rationalizing slavery in a world where you were born part of the upper caste, and slaves are said to be part of a lowborn caste that has always existed and will always exist.
But how did Ancient Greeks in particular, or Roman citizens if applicable, reconcile the institution of slavery and the idea of a slave as "living tools", with the fact that capture by a rival city might mean one's own enslavement?
Did it create any sort of moral dilemma, controversy or at least some sense of unease among citizens? Did some thinkers recommend less harsh treatment toward slaves who were born free? Was there empathy for them?
According to Robert Garland's "The Other Side of History", Greek and Romans were so used to the reality of slavery in their civilization that there didn't seem to be any sort of empathy about it. Even the best minds like Aristotle would have convinced themselves that slaves are slaves because they are lesser humans (saying they didn't "deliberate" and that their bodies were deformed). In Greece, depending on the type of slave, they would have potentially viewed them as a sort of "family member" but still would have viewed them as unquestionably inferior. Romans would free their slaves and help them become citizens, but they still seem to have a similar mentality. It's hard to say for sure if they would have feared becoming slaves.
For the Romans at least, I figure there must be some Level of worry because the Greeks became their slaves after being slave owners themselves. Then again the Roman Empire must have seemed uncrushable.
Garland has entire lectures dedicated to slavery in these civilizations. I recommend listening to them if you're curious about getting into the minds of ancient peoples. You can access them from "The Great Courses"