Why would I want to start a riot when I command a Warchief?

The Riot begins a power struggle indicated on your map by a red marker. If you go, it will create a distraction for you to get to your target, I don't think you need to draw the Warchief out in a riot, not sure. If you don't go however, you run the risk of your branded warchief or captain losing the power struggle.

If your warchief kills the other warchief or captain someone else will be promoted, not necessarily your branded captain, but generally it's one of the bodyguards. If you send a captain against a warchief and he succeeds in the power struggle, he will take the warchief's place.