Format the panel and admin date and clock in the stored application timezone

The panel snapshot now carries the admin-saved `timezone` ("" when unset);
the coordinator threads it onto the household due-today sensor, and the
Party Board and Quest Log cards format their kicker, header and dock
date and clock in it, falling back to Home Assistant's zone only when
none is stored. The admin app loads the stored zone once
(AppTimezoneProvider) and uses it for completion clocks, History day
grouping, clocks and lateness, and "today" in Schedule, History and new
task defaults; Settings publishes a saved zone app-wide at once.

Resolves review finding TZ-001 (task 7e173881).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

tests/test_api_household_clock.py

@@ -632,6 +632,24 @@ async def test_admin_saved_zone_is_authoritative_across_surfaces(
     assert delays[1] == pytest.approx(299)
 
 
+async def test_panel_snapshot_publishes_the_stored_zone(
+    household: SimpleNamespace,
+) -> None:
+    """The panel snapshot carries the stored application zone verbatim so
+    the cards format their visible date and clock in it — ``""`` until the
+    admin saves one, then exactly the saved zone."""
+    async def panel_zone() -> str:
+        response = await household.client.get(
+            "/api/v1/panel/snapshot", headers=_panel_headers()
+        )
+        assert response.status_code == 200
+        return response.json()["timezone"]
+
+    assert await panel_zone() == ""
+    await _set_zone(household, "Pacific/Auckland")
+    assert await panel_zone() == "Pacific/Auckland"
+
+
 # --- settings writes are serialised with planning and sweeping (TZ-003) ------
 #
 # A zone write racing the loop's own settings reads must never leave a