Make the stored application timezone authoritative everywhere

The kids' Quest Log again renders each instance's overdue state from
the API snapshot's `overdue` flag, which the API computes in the stored
application timezone. The browser-local overdue check from task
fd1093b5 (`_isOverdue` against `_now`) is removed; `_now` only drives
the header clock and date.

The admin settings route (PATCH /api/v1/admin/settings), the only
settings writer, now holds the missed-sweep scheduler's settings_lock
across the write and calls settings_changed, so saving `timezone` or
`day_rollover_time` re-plans a sleeping scheduler (review finding
TZ-001). An app without a scheduler just writes.

Tests: the card shows the snapshot's overdue state whatever the
browser's zone or clock says; an admin PATCH of the zone or the
rollover re-plans a sleeping scheduler; one saved zone
(Asia/Kolkata) is read identically by the panel snapshot, the admin
snapshot and the scheduler delay.

Task: 7e173881-fb99-426f-a2f0-2678439ffc39

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

frontend/src/cards/quest-log.ts

@@ -36,6 +36,7 @@ type PanelInstance = {
   window: string;
   due_time: string | null;
   state: string;
+  overdue: boolean;
   completed_at: string | null;
   on_time: boolean | null;
 };
@@ -153,17 +154,6 @@ function formatWallClock(value: string | null): string | null {
   return `${hour12}:${match[2]} ${suffix}`;
 }
 
-/** Minutes past midnight of a household ``HH:MM`` wall time, or null. */
-function wallClockMinutes(value: string | null): number | null {
-  const match = value ? /^(\d{1,2}):(\d{2})$/.exec(value.trim()) : null;
-  if (!match) {
-    return null;
-  }
-  const hours = Number(match[1]);
-  const minutes = Number(match[2]);
-  return hours > 23 || minutes > 59 ? null : hours * 60 + minutes;
-}
-
 function asString(value: unknown): string {
   return typeof value === "string" ? value.trim() : "";
 }
@@ -263,6 +253,7 @@ function toPanelInstance(value: unknown): PanelInstance | null {
     window: asString(row.window).toLowerCase(),
     due_time: asString(row.due_time) || null,
     state,
+    overdue: row.overdue === true,
     completed_at: asString(row.completed_at) || null,
     on_time: row.on_time === true ? true : row.on_time === false ? false : null,
   };
@@ -1758,20 +1749,6 @@ export class NestQuestQuestLogCard extends LitElement {
     `;
   }
 
-  /** Overdue by the panel's own clock: the browser runs on the
-      household's real local time, so an open quest turns overdue once its
-      ``due_time`` wall time has arrived there — whatever zone the API or
-      Home Assistant computed the API's ``overdue`` flag in.  With no due
-      time it is never overdue (the API's rule too).  ``_now`` is reactive
-      and ticks every 60 s, so the line flips on the minute it falls due. */
-  private _isOverdue(instance: PanelInstance): boolean {
-    const due = wallClockMinutes(instance.due_time);
-    if (instance.state !== "open" || due === null) {
-      return false;
-    }
-    return this._now.getHours() * 60 + this._now.getMinutes() >= due;
-  }
-
   private _timeZone(): string | undefined {
     return hassTimeZone(this.hass);
   }
@@ -1856,6 +1833,7 @@ export class NestQuestQuestLogCard extends LitElement {
         continue;
       }
       instance.state = "completed";
+      instance.overdue = false;
       instance.completed_at = stampedAt;
     }
     if (settled) {
@@ -2152,7 +2130,7 @@ export class NestQuestQuestLogCard extends LitElement {
       `;
     }
     const dueAt = formatWallClock(instance.due_time);
-    const meta = this._isOverdue(instance)
+    const meta = instance.overdue
       ? {
           text: dueAt ? `Overdue · due ${dueAt}` : "Overdue",
           late: true,