Did medieval people think of themselves as existing in the middle times? Of course they didn't. It's an Italo-centric idea- based on the belief that Imperial Rome represented a high point of civilisation from which there was subsequently a great falling off. The stigmatization of that post Roman period as the Dark Ages is something we owe to Petrarch- who put the phrase into print in 1330.
There was a reaction in favour of the Middle Ages around the turn of the 18th and 19th centuries- and the chivalric code and gothic architecture were thereafter recognised as the peaks of civilisation that they are.
Time flows or- if we're being really esoteric- doesn't actually exist at all. It certainly doesn't come in great chunks. And yet we still talk about The Ancient World, the Middle Ages and the Modern Era as if they had walls around them.
How hard it is to get rid of these memes.