Completely Random Events, especially with negative consequences, are just an example of a game programmer with no imagination, who is compensating for his dull game design by throwing bricks at you to keep your attention.
The "Events and Decisions" Mod for Civ V, which was based on a similar, much more elaborate mechanism in Europa Universalis IV, is an example, I think, of how to implement 'random' events: the events represented things that could happen, but each required you to make a decision as to how to handle it which in turn resulted in various negative/positive effects on your game. I would add to that a mechanism where the accumulation of decisions might lead you down a certain path in your civilization.
For instance, if most of your reactions to minor disasters (Plague in X City/Province! River Flooded villages near City X! The Heir to the Throne Insists That He Be Addressed As "The Royal Twinkie"!) are to ignore them, on the one hand your population will stop expecting you to react to them, on the other hand they may decide you are a Heartless Wretch and revolt. BUT it's Your Decisions that complicate your game, not just the Random Events themselves.
AND - as I've said before in other posts - ideally each Random Event should NEVER be entirely Negative. There have to be results that are both negative and positive, or at least negative and neutral.
Example: Volcanoes and volcanic Eruptions were a feature of - I think it was Civ IV, or maybe III - and are frequently mentioned as Something To Bring Back. Fine, but in addition to the potential of an eruption that removes population from the nearest city, or destroys Improvements in X tiles, also have the Positive Aspects of Volcanoes - the volcanic ash fertilizes the soil making it more productive, volcanic tephra is a prime material for producing waterproof concrete/cement (a Roman engineering 'secret'!) and volcanoes are sources for Obsidian, an alternative material for blades/weapons/tools if you have no access to copper/bronze/iron.
If all the consequences are negative, it will be turned off every time...