fix: switch to PostHogProvider per official React integration guide (#112)

- Install @posthog/react and wrap app with PostHogProvider
- Use VITE_PUBLIC_POSTHOG_KEY and VITE_PUBLIC_POSTHOG_HOST env vars
- Use defaults: '2026-01-30' for recommended settings
- Remove manual initAnalytics() call — Provider handles initialization
- Analytics module now checks posthog.__loaded for readiness

Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

frontend/src/main.tsx

@@ -1,36 +1,44 @@
 import "./instrument"; // Sentry must init before any other imports
-import { initAnalytics } from './lib/analytics'
-
-initAnalytics() // PostHog product analytics
 
 import { StrictMode } from 'react'
 import { createRoot } from 'react-dom/client'
 import { reactErrorHandler } from '@sentry/react'
 import { HelmetProvider } from 'react-helmet-async'
+import { PostHogProvider } from '@posthog/react'
 import { Toaster } from 'sonner'
 import './index.css'
 import App from './App'
 
+const posthogOptions = {
+  api_host: import.meta.env.VITE_PUBLIC_POSTHOG_HOST || 'https://us.i.posthog.com',
+  defaults: '2026-01-30',
+} as const
+
 createRoot(document.getElementById('root')!, {
   onUncaughtError: reactErrorHandler(),
   onCaughtError: reactErrorHandler(),
   onRecoverableError: reactErrorHandler(),
 }).render(
   <StrictMode>
-    <HelmetProvider>
-      {/* Toast notification system - theme syncs automatically via CSS custom properties */}
-      <Toaster
-        position="top-right"
-        expand={false}
-        closeButton
-        visibleToasts={3}
-        gap={8}
-        theme="dark"
-        toastOptions={{
-          className: 'sonner-toast-custom',
-        }}
-      />
-      <App />
-    </HelmetProvider>
+    <PostHogProvider
+      apiKey={import.meta.env.VITE_PUBLIC_POSTHOG_KEY || ''}
+      options={posthogOptions}
+    >
+      <HelmetProvider>
+        {/* Toast notification system - theme syncs automatically via CSS custom properties */}
+        <Toaster
+          position="top-right"
+          expand={false}
+          closeButton
+          visibleToasts={3}
+          gap={8}
+          theme="dark"
+          toastOptions={{
+            className: 'sonner-toast-custom',
+          }}
+        />
+        <App />
+      </HelmetProvider>
+    </PostHogProvider>
   </StrictMode>,
 )