Why exactly would a Byzantine nation (Anatolia + Greece at the very least) not hold a lot of influence in the world (middle ages and maybe the modern world)? I figured that a nation state like that would hold a lot of influence in the world, for several reasons: big population, strong economy, powerful navy and army and figured that Byzantium would hold a fair bit of influence in the world for those reasons.
There's no particular reason to assume it's more influential than, say, modern Turkey or Spain. It's possible and of course more fun to put them among the leaders of the modern era and the forefront of the industrial revolution, but I think the odds are they don't set the pace. And a strong economy (and strong navy specifically) require being heavily industrialized.
Obviously most countries, and specifically most eastern European areas did *not* manage to benefit relative to the pack from the modern era. The different traditions, alphabet, political system and religion probably remove them from many of the liberalizing effects of the Reformation.