fix(router): update route rune synchronously in navigate()

navigate() wrote location.hash and left router.route to the asynchronous
hashchange event. bootstrap flips app.ready in the same tick right after
navigate('/login') on an unauthenticated cold start, so for one frame the app
rendered with app.ready=true and the stale route ('lobby', from the empty hash)
— briefly mounting the lobby shell (tab bar + a doomed games.list) under the new
login screen before hashchange settled it.

Update router.route synchronously inside navigate(); the later hashchange
re-parses to the same value. This removes the boot lobby-flash (and its spurious
games.list 401) and, as the root cause, the offline e2e flake: enterLobby could
latch that transient lobby tab-bar instead of clicking through login, then the
Settings-tab click at line 44 fought the tab detaching into the login slide
(~7% of runs, both engines).

Also:
- offline.spec enterLobby: wait for the guest button and click it, instead of a
  point-in-time count() that a pre-login splash frame sampled as 0 (skipping the
  click and hanging / latching the transient).
- New e2e router.spec pins the synchronous-route property (RED on the old code,
  which returned the previous route; GREEN now) via a mock-only __router seam.

Verify: check 0 / unit 490 / build / bundle 114.3/115 / full e2e 200 / the
offline+router stress at 40x on both engines 160/160 (was 6/80 failing).

ui/e2e/offline.spec.ts

@@ -13,10 +13,12 @@ async function forceStandalone(page: Page): Promise<void> {
   });
 }
 
-// After a load or reload, land in the lobby (dismiss the login screen if it is shown).
+// After a load, land in the lobby. The mock cold-starts on the login screen (no seeded session), so
+// click through as guest. Waiting for the button (rather than a point-in-time count()) is what makes
+// this deterministic: a count() sampled during the pre-login splash frame returned 0, skipped the
+// click, and then hung — or latched a transient lobby tab-bar — instead of opening the real lobby.
 async function enterLobby(page: Page): Promise<void> {
-  const guest = page.getByRole('button', { name: /guest|гост/i });
-  if (await guest.count()) await guest.first().click();
+  await page.getByRole('button', { name: /guest|гост/i }).first().click();
   // The lobby tab bar has three tabs in a fixed order: New (0), Stats (1), Settings (2). nth() is
   // robust to locale, emoji variation selectors and the coachmark anchors (which the first-run
   // onboarding strips after it completes).