Well spoken, Camikaze.
I heartily approve of random events. They make each game unique.
For the most part, if you maintained a treasury so that you could deal with emergencies and capitalize on opportunities they had small effects and were balanced. They could help you or hurt you, but the A.I.s would be at the same risk.
Some stand out as being more game changing.
The Vedic Aryans- Early in the game you are notified that they are at your borders. It's a stack of barbarian archers. Maybe the qualifier or trigger had something to do with barbs replacing animals and the discovery of archery, I don't know. It was often a game over event.
There I am with one or two little cities, defended by one warrior each, with a worker each. My other scouts and warriors are exploring and can't return for 10 turns, and I don't have the ability to rush build defenders. So figuring I had nothing to lose, I used my fleet-footed worker to approach the stack, and try to decoy it away from my capital. They followed. I led them to the neighboring empire, which they promptly captured and converted to poorly defended barb cities. Expansion problem solved! So that event could be a game over, or it could mean the end of your dangerous neighbor, depending how you learned to improvise and adapt. Either way it's a civ-killer, but one outcome is much more satisfying, especially when you work it out for yourself.
There's probably no bigger hammer sinkhole than the Bermuda Triangle mysteriously swallowing up your invasion task force. It's incredibly frustrating. Potentially it's a game re-balancer, although a human's much more likely to plan a large scale amphibious assault. Sometimes it's only a single ship. But it's later game, you can rebuild, and not as critical as some early events.
Another big late game event was the inflation event- some economist reduces your inflation by a 4th or 5th, which could make you wildly happy, or could make one of your rivals competitive again if they made the discovery.
The discovery of Tower Shields, which gave the cover promotion free to swords ( or was it all melee ?) was probably the most pivotal one. If you were busy building wonders you had to reconsider your strategy. If an aggressive civ got it, you had to reconsider your strategy.
There were other similar promotion-conferring events, such as improved flintlocks, but coming later in the game they had less impact.
I think my favorite one is the Herbalist one. Early in the game your herbalists have discovered a new plant which they believe will confer health benefits. You decide whether to ignore the discovery, or experiment upon the populace. If you do experiment you risk population loss, with the potential benefit of permanent health boosts. Whether you choose limited or widespread experiments raises the stakes. I usually gamble big.
I think the Civ series is about decision making, and that random events introduce more decisions. I think that the "One...more...turn..." factor was a function of the suspense created, waiting to finding out the consequences of my many decisions. I think that Civ IV was the most replayable game I ever enjoyed, by far, in no small part because of random events. I think that adapting to the unexpected is a test of mental flexibility as much as playing a predictable game over and over is a test of skill.