Granted the timescales in Civ have never been realistic. But the length of time it takes to build something depends on the production in that city on a per-turn basis; it's an essential component of the game and is tied into game mechanics in a way that makes sense. Having tiles look rainy for a few turns without affecting anything else would be unnecessary, take up more processing power without adding anything to gameplay, and, yes, would be out of place with the timescale on top of that.
I could see having monsoons, and hurricanes, and volcanoes, and earthquakes, and any number of other natural disasters, occur which actually affect the game. Say, monsoons slow down units, and earthquakes destroy buildings. (Though inevitably there would be a huge outcry from the fan base about how they don't like their strategy being affected by such randomness, I would argue that in a game about history, randomness is not out of place. The tide of history often turns on events no one had any control over. We're building empires, not playing chess.) Those could certainly be depicted visually. But to just have it rain every once in awhile for no reason other than that rain looks neat? No.