fix: switch to PostHogProvider per official React integration guide

- 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/package-lock.json

@@ -13,6 +13,7 @@
         "@dnd-kit/sortable": "^10.0.0",
         "@dnd-kit/utilities": "^3.2.2",
         "@monaco-editor/react": "^4.7.0",
+        "@posthog/react": "^1.8.2",
         "@sentry/react": "^10.42.0",
         "@sentry/vite-plugin": "^5.1.1",
         "@stripe/stripe-js": "^8.7.0",
@@ -1597,6 +1598,22 @@
         "cross-spawn": "^7.0.6"
       }
     },
+    "node_modules/@posthog/react": {
+      "version": "1.8.2",
+      "resolved": "https://registry.npmjs.org/@posthog/react/-/react-1.8.2.tgz",
+      "integrity": "sha512-KzUuXIcAR8fAjU7IeDq+XfEcUTNvzgEGB381WRrFUUsu7jFTcKZZ6crx/ukHRCzTnoEuy5EJDkL7b7sJecPlCg==",
+      "license": "MIT",
+      "peerDependencies": {
+        "@types/react": ">=16.8.0",
+        "posthog-js": ">=1.257.2",
+        "react": ">=16.8.0"
+      },
+      "peerDependenciesMeta": {
+        "@types/react": {
+          "optional": true
+        }
+      }
+    },
     "node_modules/@posthog/types": {
       "version": "1.360.2",
       "resolved": "https://registry.npmjs.org/@posthog/types/-/types-1.360.2.tgz",