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.

If you send them against another warchief, you can intervene in the battle and it will be 2 uruk gangs fighting againts each other, one of them on your side. You can either use the distraction to sneak up on the enemy or snipe the unbranded uruks and let your guys soften up the warchief.

If the warchief dies, he'll be replaced by the surviving bodyguard with highest power. If you take advantage of the distraction of the riot to take out higher-level unbranded bodyguards mid-fight before taking out the warchief, you can get away with only branding 1 bodyguard before the fight.