Rewrite Quickstart guide with type-checked code examples

Rewrites the Quickstart tutorial to provide a more focused, step-by-step
introduction to building MCP Apps. The guide now uses synced code fences
that pull from `examples/quickstart/`, ensuring all code shown in the
documentation is type-checked and tested.

Key improvements:
- Clearer project setup instructions with explicit commands
- Streamlined explanations of the tool + resource registration pattern
- Added screenshot showing the completed app
- E2E test coverage validates the example works end-to-end

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

examples/quickstart/tsconfig.json

@@ -0,0 +1,19 @@
+{
+  "compilerOptions": {
+    "target": "ESNext",
+    "lib": ["ESNext", "DOM", "DOM.Iterable"],
+    "module": "ESNext",
+    "moduleResolution": "bundler",
+    "allowImportingTsExtensions": true,
+    "resolveJsonModule": true,
+    "isolatedModules": true,
+    "verbatimModuleSyntax": true,
+    "noEmit": true,
+    "strict": true,
+    "skipLibCheck": true,
+    "noUnusedLocals": true,
+    "noUnusedParameters": true,
+    "noFallthroughCasesInSwitch": true
+  },
+  "include": ["src", "server.ts", "main.ts"]
+}