Somewhat similar to how the West let liberalism take over. At one point it would have been easy to remedy things, then at a later point difficult but still possible with rolled up sleeves and lots of work, but now it is at a point where even 'conservatism' is, for the most part, a more conservative form of liberalism. Compare the average 1971 Democrat with the average 2013 Republican. The 70s Dem is far more conservative. Maybe that is what the true meaning of slow zombies is. That ignoring an easily handled threat for long enough, be it consumerism (I believe that was the target for the Romero movies????), liberalism, or simply a growing belly and pant size, eventually leads to a situation where losing that weight is almost impossible.
As for fast zombies, they are the viable ones. Immediate infection that fuels frenetic attack. Even with a gun it would be difficult taking one down unless you see it coming from some distance. Kind of like the 21 foot rule, but with teeth instead of a knife. You need to shoot it in the head since a center mass shot is useless. It only needs to break your skin with its teeth. Add two, or more, and the situation can truly be bad. The only option would be massive quarantines, followed by massive culling, since once the equilibrium shifts there is nothing else to do apart from either wait to be infected, or start dropping nuclear warheads.
However, your point about instant infection is valid. For true global spread the rate of infection has to be slow, to enable people who know they have been bitten, but do not want to be quarantined (read: liquidated) to travel and escape. A slow infection rate, coupled with the fast variant of zombies and the ease and speed of modern air travel, would lead to outbreaks in major population centers around the world.
An easy way of seeing this is how the flu spreads annually. Always from Asia, and then spreading outwards from there. That is how they are able to come up wiht a vaccine ...simply see what is brewing and cook up something for it.
Anyways, the slow zombie types from the Romero movies and the Walking Dead series could never take over the world. The moment people started becoming aware of reanimated things shuffling and shambling about biting people, they would eliminate them. They are slow, dumb, and provide ample warning with their moans. Once the initial shock passed even 9th graders would be killing them with hockey sticks!
However, if it is fast zombies then unless there is immediate quarantine and mass culling, which would even include people who are not infected, the entire world would be overrun. An apartment of fast infected could easily take over the entire block, a block of fast infected could easily take over an entire city, a city of fast infected ...the math becomes quite clear. By the time people became aware what was happening several cities around the world would already have a small but rapidly growing population of fast infected that would be more than doubling every day. Actually far more than 2X, considering that every moment the fast infected would be biting non-infected. Could be close to 100X.
Gehenna.
Now let me leave fantasy land and go do some real work. However, the same applies for virulent diseases like small pox (google a story called 'The Demon in the Freezer' to see how beating that thing was difficult, and while it is 'eradicated' from the population the virus still exists in various labs) and, ofcourse, the potential threat of a super-flu. Not the H1N1 that many FReepers stupidly mock as a fake threat, but a redux of the Spanish Flu. Their spread would be similar to that of fast zombies.