docs: comprehensive API documentation from source code

Replace existing docs with fresh documentation built entirely from source
code analysis. OpenAPI 3.1 spec as source of truth, plus human-readable
Markdown with curl examples, response samples, and workflow guides.

- OpenAPI 3.1 spec covering all 42 endpoints (validated against source)
- 7 endpoint reference docs (auth, games, sessions, picker, stats, votes, webhooks)
- WebSocket protocol documentation (auth, subscriptions, 4 event types)
- 4 guide documents (getting started, session lifecycle, voting, webhooks)
- API README with overview, auth docs, and quick reference table
- Old docs archived to docs/archive/

Made-with: Cursor

docs/api/endpoints/stats.md

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+# Stats Endpoints
+
+Aggregate statistics about the game library, sessions, and popularity.
+
+## Endpoint Summary
+
+| Method | Path | Auth | Description |
+|--------|------|------|-------------|
+| GET | `/api/stats` | No | Get aggregate statistics |
+
+---
+
+## GET /api/stats
+
+Return aggregate statistics: game counts, pack count, session counts, total games played, most-played games, and top-rated games.
+
+### Authentication
+
+None.
+
+### Response
+
+**200 OK**
+
+```json
+{
+  "games": { "count": 89 },
+  "gamesEnabled": { "count": 75 },
+  "packs": { "count": 9 },
+  "sessions": { "count": 12 },
+  "activeSessions": { "count": 1 },
+  "totalGamesPlayed": { "count": 156 },
+  "mostPlayedGames": [
+    {
+      "id": 42,
+      "title": "Quiplash 3",
+      "pack_name": "Jackbox Party Pack 7",
+      "play_count": 15,
+      "popularity_score": 8,
+      "upvotes": 10,
+      "downvotes": 2
+    }
+  ],
+  "topRatedGames": [
+    {
+      "id": 42,
+      "title": "Quiplash 3",
+      "pack_name": "Jackbox Party Pack 7",
+      "play_count": 15,
+      "popularity_score": 8,
+      "upvotes": 10,
+      "downvotes": 2
+    }
+  ]
+}
+```
+
+| Field | Description |
+|-------|-------------|
+| games.count | Total number of games in the library |
+| gamesEnabled.count | Number of enabled games |
+| packs.count | Number of distinct packs |
+| sessions.count | Total sessions (all time) |
+| activeSessions.count | Sessions with `is_active = 1` |
+| totalGamesPlayed.count | Total game plays across all sessions |
+| mostPlayedGames | Top 10 games by `play_count` DESC (only games with `play_count` > 0) |
+| topRatedGames | Top 10 games by `popularity_score` DESC (only games with `popularity_score` > 0) |
+
+### Error Responses
+
+| Status | Body | When |
+|--------|------|------|
+| 500 | `{ "error": "..." }` | Server error |
+
+### Example
+
+```bash
+curl "http://localhost:5000/api/stats"
+```