Cycling in the rain in the city your tyres can slip on oil - unpleasant results even at starting speed. The oily puddles also stain clothes, which probably aren't cycling gear on a short 15 minute ride. My last job was a Chiswick to Tower Hill ride, gear up and get dirty, but the worst part was the state of the toilets we had to change in.
For short city trips, you want to have an old cheap outdoor bike (otherwise it might get stolen anyway). I don't see why you have to clean those. (I clean my bike once every 6 months, tops, and I ride it every day. The dirt is from the dirt or forest tracks I have to cross once in a while.)