A renaissance is when the best of the "old days" is brought back. When northern Italy started bringing back the Roman and Greek sculpture and writing, they didn't bring back gladiatorial games, did they? They just chose the bits they liked from the "old days", and chucked out the rest.(He went on to say some other, ridiculous things, but that's neither here nor there.)
I think it's useful to point out that The Renaissance wasn't really "bringing back the Roman and Greek sculpture and writing". It had the effect of bringing back classical style and an interest in classical texts, but that's not really what The Renaissance was. It wasn't just a fashion craze.
Medieval Europe had lost a lot of classical knowledge. They didn't know how to build roads the Roman way, roads that would last. They didn't know how to build aqueducts. They had some pretty impressive engineering skills when it came to building castles, but they didn't know how to build buildings the way the Romans or Greeks did. They were preserving some philosophical works in Latin, but they'd lost most of the technical works, the scientific knowledge.
As James Burke put it in either Connections or The Day the Universe Changed, the medieval Europeans, and the medieval Italians in particular, were surrounded by old stuff they didn't know how to duplicate.
What happened was that some Italian idle rich found out the Islamic world had preserved some Greek and Roman technical texts, and had even added some of their own findings. They paid for translations into Latin or even Italian and started distributing books that told how to build massive works or use perspective in art. Suddenly, by studying old texts, they'd discovered the key to doing stuff the classical way. They could duplicate it, or even take it in new directions.
The Old School Renaissance works in much the same way. It's not "playing with old rules"; some people never stopped doing that. It's not like medieval history, where people had actually lost old texts; we still had some OD&D, classic D&D, and AD&D books (in languages we could read) floating around. It's not about duplicating stuff that existed before, even though there's been some of that. It's about rediscovering *why* people did things that way, and restating those insights in a clear way, and using those insights to improve the way we play old school games, or even to create completely new games based on those insights.
It's a true Renaissance, a recovery of lost knowledge.