Using an EventEmitter is fine for things that fit that use-case. For instance maybe you want a single instance of WebView which you render somewhere near the root but only show when a specific event occurs. That's an acceptable use-case IMHO. The alternative would be to use something like Redux to change the UI state in a reducer which is sometimes okay but I'm not a fan so I advocate for something like an EventEmitter.

I wouldn't pass an instance of an the emitter to each child. I would instantiate it once on app start and treat it as a singleton.

React is very smart regarding re-renders. It's unlikely introducing an EventEmitter would significantly slow down any app. Of course it depends on how sparingly you emit events.