Wires the Matchmaking menu section's Find Match button to the real
queue:join/queue:leave flow: a searching state with cancel, the
queue:banned countdown (banBanner.ts) with Find Match hidden until it
expires, a brief assigned-mode reveal on match:found, and the handoff
into MultiplayerMatch's two-phase init once round:start is known.
After a match ends (round:reveal + result dismissal, local quit, or
opponent quit via room:closed) the flow returns to the matchmaking
menu without re-queuing.
Also adds a shared GameClient singleton (net/client.ts) so every UI
flow that talks to the server reuses one connection.
Note: backend/src/matchmaking/queue.js actually sends round:start
before match:found (the reverse of PROTOCOL.md's documented order),
discovered via live two-tab testing against the real backend;
matchmakingFlow.ts tracks both messages independently so either
arrival order works.
Gabriel Franco
8bb0c79659