feat(mgmt/web): show the running game, and who decides when it ends

The host knows what it launched and, after a disconnect, that it is counting
down to closing it. None of that was visible: the console showed a stream with
no game, and a game on its way to being closed was something you found out about
afterwards.

The Dashboard gains a running-game card above the session card — box art matched
against the catalog it already fetches, so no new endpoint and no request on the
2 s status poll. "End now" means the two different things the row's state
implies: a live game ends by stopping its session (what then happens to the game
follows the policy — stopping a session is not licence to close a game), while
one already waiting out its reconnect window has no session left to stop and is
ended directly.

The settings card sits next to the display keep-alive policy because that is the
same question one step out: keep-alive decides how long a *display* outlives a
disconnect, this decides whether the *game* does. The copy says plainly what
`always` costs, that a drop is not someone pressing Stop, and that a display kept
forever is unaffected either way — the precedence rule that would otherwise
surprise someone. Where a build enforces nothing (macOS, no launch path) the
controls are shown disabled rather than hidden: "does nothing here" is
information.

Also fixes the story fixtures, which had gone stale against the `games[]` field
Phase 1 added, and adds a story for the state the card exists for — a game whose
client walked away.

web: build + tsc clean, biome-formatted; en/de messages complete.

web/src/stories/Dashboard.stories.tsx

@@ -1,6 +1,6 @@
 import type { Meta, StoryObj } from "@storybook/react-vite";
 import { DashboardView } from "@/sections/Dashboard/view";
-import { statusActive, statusIdle } from "./lib/fixtures";
+import { statusActive, statusGrace, statusIdle } from "./lib/fixtures";
 
 const meta = {
 	title: "Pages/Dashboard",
@@ -8,8 +8,10 @@ const meta = {
 	args: {
 		onStopSession: () => {},
 		onRequestIdr: () => {},
+		onEndGame: () => {},
 		isStopping: false,
 		isRequestingIdr: false,
+		isEndingGame: false,
 	},
 } satisfies Meta<typeof DashboardView>;
 
@@ -23,3 +25,8 @@ export const ActiveSession: Story = {
 export const Idle: Story = {
 	args: { status: { data: statusIdle, isLoading: false, error: null } },
 };
+
+/** A game whose client vanished: the host closes it when the countdown runs out. */
+export const GameWaitingForItsClient: Story = {
+	args: { status: { data: statusGrace, isLoading: false, error: null } },
+};