feat: add comprehensive auth debugging logs

[KyleTryon]

Summary

Add detailed logging for cross-domain authentication debugging to help diagnose staging login redirect issues.

Changes

Frontend Logging

• auth-client.ts: Log Better Auth client configuration (baseURL, environment)
• useAuth.ts:
  • Log login flow steps (type detection, API result, navigation)
  • Log post-login navigation (session fetch, cookies, router invalidation)
• __root.tsx: Log root beforeLoad session fetch results

Backend Logging

• better-auth.ts: Log Better Auth configuration (cookie domain, CORS origins, cross-subdomain settings)
• app.ts:
  • Log CORS middleware configuration
  • Log incoming auth requests (method, path, origin)
  • Log Set-Cookie headers in auth responses

Why This Helps

When testing on staging, you'll see in the browser console:

• What baseURL the auth client is using
• Login flow progression
• Session fetch results
• Cookie state
• Navigation attempts

And in Cloudflare Workers logs:

• Better Auth configuration (especially COOKIE_DOMAIN)
• CORS origin configuration
• Auth requests and their origins
• Cookie headers being set

This will immediately show if:

1. Frontend is pointing to the wrong API URL
2. Cookies are being set with the wrong domain
3. CORS is blocking requests
4. Session fetch is failing after login

Testing

Deploy to staging and check:

1. Browser DevTools Console for [Auth] and [Root] logs
2. Cloudflare Workers logs for 🔧 and [Auth] logs