Exactly! It's funny, because even in your Beijing maps, they still look strongly old-world to me, with streets that meander and curve. But now that you point it out, they curve somewhat less strongly than streets in the very old city in your other image. They also seem to have a preference, more or less, for the cardinal directions, even if they wiggle a bit.
It's interesting you mention Rome, because I was looking at the Indian city of New Dheli just recently. New Dheli was created by the British, I believe, and its layout struck me as strongly reminiscent of Washington, D.C., which was created in a consciously neo-classical style, imitating the old Roman empire.
I wonder what makes really old streets curve so much? I've thought about this before. They must have started out as footpaths or cart roads. But if you travel by foot over foot-crossable terrain, interested in getting from point A to point B, wouldn't you walk in a straight line? So what makes long footpaths curve when there's nothing else around yet? Is it just a matter of going around hills? One way to test that would be to find an ancient city on absolutely flat land (with no rivers, because then the paths would obviously have to respect the winding course of the water). Somewhere like, say, Midland, Texas. But I suppose that, in order to have water nearby, you need rivers and hills around, because hills imply valleys, which is where you'll find water, barring technology for pumping deep aquifers.
Maybe they just grew like a mycetozoa fungus.
I guess I should say at least one thing that's on-topic in this post, so while we're on the subject of topology, wouldn't it be amazing if the streets actually reflected topological features? Now that would be simulation!