Yes, Civ3 was a rushed game due to the falling out between Sid and Brian Reynolds. Soren was brought in at the last minute and the game rebuilt from scratch in 18 months(?). Reynolds wanted an RTS game (recall this was a time when RTS on PC had a huge cache and was the hot thing to do- that game ended up being rise of nations)
Years on, it is hard to argue who was right here, and I think Sid's instinct to insist the franchise stick to TBS may be one of his biggest and greatest contribution to the franchise as the absentee lead designer. It was a positive decision.
Also I think you're underselling Civ3 by a bit here. Not only did it provide the base for Civ4, Civ3 fundamentally changed how Civ games were played that has shaped the 2 entries following it.
Before Civ3, AI stuck to a set build pattern. They are thus unable to value buildings and units correctly. When stats changed in mods, patches or if the situation on the ground changed it still used the same old build order. Soren Introduced a valuation table where each stat is assigned a value, and thus relative importance can be weighted and measured by the AI. This is perhaps the single most significant change, without it, the modern Civ could not be what they are.
Before Civ3, diplomacy & trades was an afterthought. Civ3 introduced the trade table with rational valuations by the AI. I personally rank this as my most important feature, because it made Civ games less about the dichotomy between peaceful tree huggers and blooth thirsty warmongers, but introduced politics to the game. My
Machiavellian Doctrine was written around power maximiation via influence using the game's systems to create de-facto vassals, city-state clients and the like, years before Civ5.
Before Civ3, civ games were about patching together a collection of city-states settled or conquered by the player into an abstract unit called an empire. As you noted, Civ3 globalized unit support but also created 'cultural borders'. Civ3's contribution is the introduction of the empire and nation state as concepts that the game AI could understand. Most importantly it shifted the synergy of the constituent parts of your empire into the whole. The whole idea of 'small' national wonders play into this. The idea of using cities to acquire resources for the national/emperial interests became the norm.
Before Civ3, resources were something of a trade/shield/food modifier. After civ3, we send millions to their deaths for one more source of X.
If anything, I should be mad at the constanty harping about Civ4 by detractors of the current game. Civ3 is so undervalued as an entry and its influence so important I would rather compare Civ5 to that game.