This mode is much more familiar to multiplayer gamers everywhere as this is how the vast majority of multiplayer games are run.
By keeping the rendering local, we remove some of the latency. We also have the advantage of hosting a local executable, where we can use local processor cycles to aid in hiding latency. Most multiplayer games use a variety of tricks to cover up latency, the major one I’m familiar with involves running a local simulation of the world physics, with the server arbitrating any conflicts. This allows the game to feel much more responsive and makes use of our local processing power.
A key point to remember is that in the case of remote-server cloud-gaming a multiplayer game, this entire process is happening on a remote computer, so you get all the latency of the cloud gaming service, plus the latency of a normal multiplayer game. Not an optimal scenario.