Riot's side of the argument would be something like: how do you think Nike would feel if Tiger Woods decided to switch to Titleist balls next week despite having a contract that pays him to use Nike golf balls?
This seems more like Nike getting mad that Tiger took his son to a Putt Putt that didn't use Nike balls. He isn't really contradicting his endorsement by using the balls that Putt Putt gives him to play miniature golf with his son, and they aren't really contradicting their position as League of Legends pros by playing other games during their downtime.
I think you've poisoned this analogy a bit with the putt putt company's semi-requirement and the difference between what someone does in their private life and in public as a performer.
If Swifty goes to Blizzcon and is forced to use a competitor's mouse to play the new xpac alpha I don't think Razer complains but if it gets out that he's using a Logitech G600 at home based on his stream he's going to lose that endorsement.
Circling back to Tiger, it'd be more like Tiger Woods scheduling his own live event, Tiger and Son Putt Putt Happy Time Hour, and he chose to use Titleist Putt Putt Balls.
These people aren't prevented from playing other games, just from publicly streaming those games. As sponsored spokespeople for Game X I don't find this an egregious requirement and if Riot isn't paying enough to make this requirement worth honoring then the players should tell balk at the terms.
It's a shame to limit people's personal pursuits