fix: survive host modal close during listed-lobby auto-start (#4514)

Follow-up to the listed-lobby auto-start feature (`2a82b01e3`, on main).

## Bug

When a listed lobby auto-started (and in some paths on manual start),
the game began server-side but a player got ejected back to the menu
before it loaded.

Root cause: `BaseModal.close()` navigates via `showPage()`, which
**force-closes whichever page-modal is currently visible**. During the
game-start transition that is the *other* lobby modal — closing the join
modal first cascade-closed the host's modal with `leaveLobbyOnClose`
still armed (host disconnected → host-left teardown killed the game);
reversing the order just moved the victim to the joiner. No close order
fixes both sides.

## Fix

- **Disarm both lobby modals before closing either** in the prestart
transition (`disarmLeaveOnClose()` on `HostLobbyModal` and
`JoinLobbyModal`), so no cascade order can disconnect anyone mid
game-start.
- Both modals re-arm `leaveLobbyOnClose` in `onOpen` instead of
`onClose`'s state reset, so once disarmed no later cascade close can
re-arm it until the modal is reopened.
- `HostLobbyModal.closeWithoutLeaving()` (pattern `JoinLobbyModal`
already used), called explicitly instead of the generic modal-close
loop.
- **Server hardening:** `handleClientDisconnect` bails on `hasStarted()`
(which includes prestart) instead of raw `_hasStarted`, so host socket
churn during the lobby → game transition can never tear down a starting
game regardless of client behavior. Regression test included.

## Verification

Headless end-to-end (two browser contexts, host + joiner, `leave-lobby`
stack-trace tripwire armed on both pages):

- **Manual start:** host and joiner both receive prestart/start and
enter the game; zero `leave-lobby` dispatches.
- **Auto-start:** same, with nobody interacting after listing.

Also confirmed manually in a real browser. The auto-start deadline was
temporarily 30s during testing and is restored to 5 minutes in the final
commit. Full suite green (1843 + 161), tsc/eslint/prettier clean.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

---------

Co-authored-by: Claude Fable 5 <noreply@anthropic.com>

tests/server/HostedLobbyListing.test.ts

@@ -332,6 +332,23 @@ describe("host-left lobby teardown", () => {
     (game as any)._hasEnded = true;
     expect(game.joinClient({} as any)).toBe("rejected");
   });
+
+  it("does not tear down when the host disconnects during prestart", () => {
+    // During the lobby -> game transition the host modal closes and sockets
+    // churn; a starting game (e.g. listed-lobby auto-start) must survive it.
+    const gm = new GameManager(mockLogger);
+    const game = gm.createGame("g-prestart", undefined, CREATOR)!;
+    game.setListed(true);
+
+    const hostWs = fakeWs();
+    game.joinClient(makeClient("host", CREATOR, hostWs));
+    (game as any)._hasPrestarted = true;
+
+    hostWs.emit("close");
+
+    expect((game as any)._hasEnded).toBe(false);
+    expect(game.phase()).not.toBe(GamePhase.Finished);
+  });
 });
 
 describe("listed lobby host powers", () => {