240hz is butter smooth. But if there's a 200ms delay, then you will still lose before you even see the enemy. The problem here is buffering. Stadia can produce 60hz video, but that doesn't mean you will see it immediately. It first needs to be encoded, buffered, transmitted, buffered, decoded, buffered, sent to the screen, buffered, refreshed, and then you'll see it. And unless people find a way to increase the speed of light, the transmission will always take time proportional to the distance between Stadia's servers and you.
Speed of light is not a big deal as long as you don't run the datacenter on the other side of the planet. Video encoding is a much bigger source of latency. The lower the latency the higher the bitrate for the same quality. Since there is an upperbound for the bitrate the cost of encoding the video will dominate.
The largest straight line distance across the US is 2802 miles (Florida to Washington), according to the internet. That's 15 light milliseconds in a vacuum. That's roughly a frame of delay at 60fps. You need to multiply by another 1.5 to account for the speed of light in fiber, and fiber doesn't go in straight lines, but at the same time datacenters and users are unlikely to be literally as far apart as possible, so let's pretend those cancel out.
You don't need to be across the world for it to matter, just across the continent.
Yes, and no one should try to play competitive games across a continent. Competitions must be regional. Internet latency is great within a few hundred miles.
This isn't a question of where the players are relative to eachother, this is a question of where the players are relative to the datacenter with tons of expensive GPUs. Also for many games where that datacenter is relative to the game servers.