I'll start... I've hinted at these in other posts, but mostly as an afterthought, and after agreeing on things with a lot of other brilliant people.
1: DIVIDE EMPIRES INTO PROVINCES
Most nations (even the United States, for simulation purposes) would have only a few provinces. California, New York, Texas, Southern US, Midwestern US, West Coast US. We're talking about 5 or 6 maximum provinces.
The incentive to have a province would be a way to reduce micromanagement -- turning your eastern border into a war machine, while your central province encourages economic growth. But a gameplay incentive is not enough, since some will still just treat their empire like a heap of micromanaged cities. Such is their choice. But having them build a provincial palace (like a forbidden palace, but limited to 5 beyond your main palace) would reduce corruption and pave the way for other ways to simplify the game.
2: EXPEDITE DOMINATION WITH SURRENDER
Rather than having to take out every city within an empire, micromanaging unit after unit, chasing down that last settler who found a city between some mountains and a desert, AI civs (or even human civs through the AI) surrender under certain circumstances, such as capture the king.
SURRENDER AND PROVINCES
By taking out a provincial palace, that province becomes yours. It will be the palace of the former civ, and won't affect the limit of palaces you can build. Essentially, all the provincial cities surrender. An immediate question is what happens to the troops in the provincial cities -- I won't just make up an answer at this point. ... I think this can be resolved though, I just won't try (yet).
Of course you can still do it the old fashioned way, and go city by city, taking the provincial palace last. But such would be a strategic choice.
ALTERNATIVE SOLUTION: MONUMENTS
Your military advisor, if he has appropriate intelligence, would tell you: "If we take out New York, Washington, and Philadelphia, the Union will certainly surrender!" Otherwise it's somewhat of a guessing game.
And if you wanted to give greater control back to the (potential) victim, let them determine the three "target" cities with some benefit. I suggest 3 national monuments. This would be different from provincial palaces in that they would have nothing to do with geography, strictly your favorite cities who deserve/need monuments.
I can't withhold my desire to re-complicate things a bit, but I'd also like to see said nations half collapse instead of entirely collapse

E.g.: now the west coast of the Roman Empire will fight, and the east coast of the Roman Empire will remain occupied... with a potential of the West Roman Empire reclaiming its territory if it reclaims all three of said monuments.
You could even add a dimension of razing a monument upon capture, to complicate the efforts of retaking the empire. Already, though, this asymmetry and number of choices is probably much more complicated than a provincial solution.
3: SIMPLIFY POPULATION CONTROL
Rather than dealing with individual population heads and balancing the math of unhappy to happy citizens, each city has a small "meter" or something beside it, going from green to yellow to orange to red. This indicates the mood of the citizens. You pacify them, as usual, through buildings and such.
Instead of having to manage citizen mood at a municipal level, it is a provincial (or even national) phenomenon. The six cities in your western province dip below a certain level -- 2 yellow, 2 orange, 2 red; or 4 red, 2 green -- and that province begins resisting.
You thus have to spend more on luxuries in that province. A solution available off a few clicks, and something you can see coming by looking at the map... instead of predicting whether a resistence will happen when another population head appears.
(And speaking of resistence, I can't resist the urge to suggest that 2 adjacent revolting provinces should cause a civil war, a new civilization to appear as an twin of your former empire. You pick which region you want to control -- North or South, East or West, Mainland or Island.)
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Anyway, those would be the first three that come to mind from other posts, elaborating on ideas that have been thrown out.
I think an obvious theme is that in creating a new concept to simplify an old concept, you make way for a new complexity to emerge from the new concept (e.g.: civil war, provincial management). The hope is that the overall complexity of the new game stays in the same ballpark as the old game.
Anyone else have ideas of how to simplify Civ 3?