Not a terrible idea in the least, but I don't think it would really solve the problem. People who are toxic in chat will not stop because one person muted them; more than a few of them want to vent and be the loudest person in the game, and the one mute won't do anything.
I also see it having the opposite effect, in some extreme cases, where if a toxic player (especially if a bot lane player) knows the other person muted them, they might do actions in game to get their attention, such as feeding, watching their partner die and walk away, things like that. I doubt that would be the norm, but I think it could bring about a few cases like that. If a player is toxic, they will find a way of being toxic, muted or not.
But it's absolutely not a terrible idea. Just not sure if it would really help. Good thought though.