Scale wise, from the served side, it these are inactive TCP sessions at the end of the day. There isn't a serious scalability problem that doesn't exist with beacons, except that you might need to create some new software to support this.

A persistent connection when a device is awake, as long as you're not sending data over it, will be nothing compared to the screen power draw.

On the other hand, most phones will aggressively unload your app when the screen goes off or when its not in focus thereby killing your socket.

I don't think there are actually many downsides to this as long as (afaict this is the case) websockets are just HTTP layer sockets downgraded to plain TCP.