perf: load calendar events instantly, sync CalDAV in background

Split loadEvents into two functions: loadEvents (instant DB read) and
syncAndReload (background CalDAV sync + reload). Events now appear
immediately when switching to the Calendar tab instead of waiting for
the CalDAV sync to complete.

CLAUDE.md

@@ -405,7 +405,7 @@ CalDAV sync with external calendar servers (e.g., Radicale) using `tsdav` and `i
 **Sync Triggers (client-side via `CaldavConfigService.sync()`):**
 - **Login** (`login.tsx`): After successful authentication
 - **Auto-login** (`AuthGuard.tsx`): After `loadStoredUser()` if authenticated
-- **Calendar timer** (`calendar.tsx`): Every 10s while Calendar tab is focused, via `setInterval` in `useFocusEffect`
+- **Calendar timer** (`calendar.tsx`): Events load instantly from DB on focus (`loadEvents`), CalDAV sync runs in background (`syncAndReload`) and reloads events after. Repeats every 10s via `setInterval`
 - **Sync button** (`settings.tsx`): Manual trigger in CaldavSettings
 
 **Lazy sync (server-side in ChatService):**
@@ -618,7 +618,7 @@ NODE_ENV=development        # development = pretty logs, production = JSON
   - Orange dot indicator for days with events
   - Tap-to-open modal overlay showing EventCards for selected day
   - Supports events from adjacent months visible in grid
-  - Uses `useFocusEffect` for automatic reload on tab focus with periodic CalDAV sync (10s interval while focused)
+  - Events load instantly from local DB on tab focus, CalDAV sync runs non-blocking in background (`syncAndReload`) with 10s interval
   - DeleteEventModal integration for recurring event deletion with three modes
   - EventOverlay hides when DeleteEventModal is open (fixes modal stacking on web)
 - Chat screen fully functional with FlashList, message sending, and event confirm/reject