The problem is that it completely ignored military power, meaning a military campaign was completely devolved into city attack and city defense. In effect the way culture worked in Civ 4 made military and espionage all but redundant because it effectively 'abstracted' the conquest of territory (and cities in some cases) down to who had the most theaters, temples, etc.
If it was to represent dissent, why did it work better than espionage? If it was to represent influence, why was military might ignored?
It is IMO quite logical that you could not use solely military power to conquer territories in CIV. You can rob, pillage and burn, but you can not exploit natural resources by military units. You need influence (=culture) which is essentially non-military (and city-centered).
The notion that military power alone can bring control of resources (territory) is proven time and time again to be wrong. Latest Middle Eastern history is but the latest example. It is influence that counts, and it is not military in its essence. You won't get oil by fortifying a, say, Navy SEAL unit on top of an oil well. You might get it by projecting cultural notions like representation, capitalism, etc. It is just the side effect of modernity that America's influence is projected through other buildings besides theatres and temples.
However, Alexander the Great's example comes to mind when it comes to temples and theatres. He tried to control the lands he conquered by building these very seats of Greek culture.
As an intermediate example, take Soviet occupation of Eastern Europe. Once again, in eerily CIV-like terms, having retaken the cities constituting what would become their satellites (by military strength, no doubt), they kept them relatively "happy" through garrisoning them with military units (enabled by a certain novel type of Hereditary Politburo Rule) while constantly pumping out all sorts of cultural buildings producing propaganda-culture. And the loss of Soviet influence over Eastern Europe which anniversary we had a happy opportunity of observing last year came about not through military strength but through what may best be described as culture. Which had been building up until certain cities flipped (and yes, not without the help or some culture-planting and revolt-inciting Espionage missions).
Which, by the way, might have been a perfect non-military and non-city-centered way of flipping tiles, had Espionage made it into CV.