More particularly, I know the pyramids were thousands of years old at the time of the peak of Roman power. But nonetheless did educated people in Roman times or earlier have a sense that they were relatively close to the darkness of prehistory? Or did that seem as far away to them as it does to us today?
Maybe it's me, but your question seems strange. Relatively recently, you say. Relative to what? Modern times? No classical society looked to the year 2015 and thought "well, we're closer to the invention of writing (= the end of prehistory) than they are" which wouldn't even be true considering writing existed already for some 3000 years before the classical era. Do you look to the year 252525 and think "my, we're practically cavemen?"
So, no. No classical society looked back and thought itself close to prehistory because no one looked to the future for comparison. Furthermore there is less time between us and classical times than there is between classical times and prehistorical society (c. 3200 BCE), not to mention the beginning of agriculture and the first cities (c. 4500 BCE).
However, I do want to say this: beginning with the first cities, a lot of societies in Mesopotamia looked back on previous eras as better than their current time. Read Gilgamesh, for example. To them, the current time was a time of great change and social upheaval and uncertainty. The past was almost always glorified and seen as better than the future.
Tl;dr: the classical times are closer to us than to prehistory. Hardly 'recently emerged'. Those that did look to prehistory as something they 'emerged from' looked at it as a 'golden age', rather than as an era of 'darkness'.