feat(demo): list the demo's sign-ins on /beta and link into it per game

The public demo (demo.runicgateway.com) is live, so the site's demo slot
can be switched on. Two decisions of record, PLAN.md §15:

- D82: /beta reads the demo's GET /api/v1/public/settings demo.logins on
  the server, caches it for a minute, and lists the published sign-ins
  with the next wipe. If the demo does not answer, /beta links to its
  sign-in page instead. Gate two reads as in place.
- D83: a deep link into a game module's pages carries data-demo-game, and
  applyBrand.mjs fills it only for games in the new mounted demoGames
  list. The Rust features gain links into /rust and /rust/servers/demo.
  UO's stay hidden on a Rust demo.

The public-demo not-built entry is removed, because its resolvedBy was met.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01E14m6SuuY6i1vASFeGDBeY

src/data/notBuilt.mjs

@@ -117,23 +117,9 @@ export const notBuilt = [
     resolvedBy: 'A game server that runs there.',
     link: { href: `${GITEA}/docs/src/branch/main/installer/INSTALL.md`, label: 'The operator guide' },
   },
-  {
-    id: 'public-demo',
-    // Phase 5 added `app` and `beta`, and that is not tidying. D27 makes the demo the
-    // deployment a beta tester connects to, so on those two pages this stopped being a
-    // thing the site lacks and became the thing the beta is waiting for. An absence that
-    // blocks a call to action has to be on the page carrying that call to action.
-    scope: ['features', 'app', 'beta'],
-    title: 'A public demo you can click through',
-    body:
-      'Planned and out of scope today: a virtual machine running the whole stack including ' +
-      'a game server, with settings locked down and an hourly reset. Until it exists this ' +
-      'site does not link to one, and there is no screenshot here of something that is not ' +
-      'running somewhere.',
-    resolvedBy:
-      'The machine being stood up. The site is already built to gain it by way of one line ' +
-      'in a configuration file, rather than a rebuild.',
-  },
+  // `public-demo` stood here until 2026-10-28. Its `resolvedBy` was met when
+  // demo.runicgateway.com went up (docs/website/DEMO_MODE.md), and the site reaches it through
+  // the mounted `demoUrl`, as the entry promised (D82, D83).
 
   /* ---------------------------------------------------------------------------------------
      THE ANDROID CLIENT (phase 5)