I think the "break the game" things are not that frequent. And I think civ is a better game when you add the **** happens factor. Just my opinion of course.
Over the course of this discussion I think I now appreciate the other side of this argument a bit more due to the insightful responses from others.
If adding a random element I'd like to see more actual strategy applied to them. "you lost this tile improvement rebuild it" isn't strategy; it's tedium. Forge burning and such are similar too; the cost in gold is so comparably cheap that you'd virtually always want to rebuild a burned building if you were willing to invest in it in the first place (IE not much strategy when you "if this" and can instantly conclude "do that" and be right consistently). As binary research is generally optimal anyway, you'll always have the money if you're not being lazy also.
Quests are a valid element. So are events that happen under more consistently discernible circumstances and can be planned to avoid, get, or compensate for. This is a strategy game, but like vassal states, AP, and most other expansion features events is largely a thrown-together package that could have made gameplay SO much more dynamic than they currently do. They'd still be bad in HoF because of "replay = skill" factor and in XOTM, but they'd be a lot more fun and more competitively valid as opponents might even seek to intercept good ones/cause risks of bad ones.
But nope. Most of these "fun" events are just something happening that you
1) have minimal-to-no control over (random doesn't mean it has to preclude strategy!)
2) are largely minor and only affect timings by a turn or two, and if you're ahead by more than that wind up being non-factors or
3) potentially game-breakingly good (free golden ages, massed march) or terrible (bermuda triangle, vedic aryans)...and even these are frequently unprepared random draws.
This is why I temper excitement over the civ V expansion; I still remember that YOU having a vassal can actually make your opponent think HE is doing better in the war than if you didn't have that vassal...even if the vassal has killed more of its units than it lost.
The same company that made choices like that to put into the game is giving us an expansion in a game that still doesn't work...and that fact escapes a large portion of the community
.