I don't really like the 1000 years method used. I'd much rather it be, say, 200 turns or less, maybe with the option of doubling multiple times.
Anything built past 1100 AD never gets to be considered a piece of "history" like other culture buildings. It's quite possible that some RL Americans structures have reached their "double" time already.
The faster ideas spread, the faster culture spreads! Modern culture exists in many forms which streak across the world at blinding speeds, like internet memes.
I don't think the Civ4 game designers properly appreciate modern culture and just assume old = cultural.
Well, they did include the whopping +50% bonus from broadcasting towers. And, for RL American culture, they have Hollywood, Rock n' Roll, and Broadway, which actually double everything. (So it also doubles, in a way)
As an American myself, I don't think Americans quite realize what it's like to be living in a place that's been around for centuries. The American identity is fresh in comparison.
TL;DR: The internet can be a vehicle for culture, but just carrying it around doesn't necessarily give it cultural depth.
It's true that culture exists in many forms. That's just the thing about the internet and memes, it usually comes in one form. There's not a lot of variance. Everybody knows the same memes. It's like having 50 copies of a book vs 1000 copies of a book. Just because there are more, doesn't necessarily mean it's "more culture." (Not to say the internet isn't cultural — it definitely can and does produce that rich variety in some areas. It's too soon to say how much.)
It needs to have multiple dimensions. Internet memes are usually rather one-sided. They're usually comedic, or sarcastic, cynical, sometimes emotional. But they don't really undergo the transformation that other symbols do. European thought has gone from Medieval chivalry to Rationalism to Romanticism to the "ehh-Romanticists" (Heine) to Realism to Surrealism... All referencing each other, and referencing the same pool of symbols and landmarks. So the original symbol takes different dimensions. Memes...they just disappear when they lose popularity. It's not about how fast ideas spread—it's about how fast they change and evolve, and what meaningful impacts they leave with every permutation.