I’m not referring to more than one machine when I say client 0 is also the server. Nor am I saying that multiplayer has to be networked.
Mario Kart, Two buds on a couch, split screen. Player 1 is client 0 and server, Player 2 is client 1.
Server is not synonymous with physical hardware. It most often refers to software that is the authority of the game simulation.
Game Mode (GM) only resides on the server. GM initializes Game State (GS). The GS initializes the Player State.
If there’s no server you have no Game Mode, Game State, Player State. Much Less Game Session in which other players can join. Join doesn’t exclusively mean over a network.