Add logging for request duration and response details in 2c

[conico974]

Implement logging to capture request duration and response details, enhancing observability.
With this PR we'll get data about the request duration, status code, if it's a cache hit or miss, and the requested route (Not the url, the nextjs route)

[linear]

RND-7918 Trace request duration and status code in 2c

[changeset-bot]

⚠️ No Changeset found

Latest commit: a003093

Merging this PR will not cause a version bump for any packages. If these changes should not result in a new version, you're good to go. If these changes should result in a version bump, you need to add a changeset.

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[github-actions]

Summary of the deployments:

Version	URL	Status
Vercel	https://gitbook-v2-nxnvdrfb5-gitbook.vercel.app	✅
Cloudflare	https://6b295ed5-gitbook-open-v2-preview.gitbook.workers.dev	✅

Test content

Site	2v	2c
GitBook	https://gitbook-v2-nxnvdrfb5-gitbook.vercel.app/url/gitbook.com/docs	https://6b295ed5-gitbook-open-v2-preview.gitbook.workers.dev/url/gitbook.com/docs
E2E	https://gitbook-v2-nxnvdrfb5-gitbook.vercel.app/url/gitbook.gitbook.io/test-gitbook-open	https://6b295ed5-gitbook-open-v2-preview.gitbook.workers.dev/url/gitbook.gitbook.io/test-gitbook-open

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Build	Status	Details	Updated (UTC)
customers-v2 (Inspect)	✅ No changes detected	-	Aug 22, 2025, 9:08 AM
v2-cloudflare (Inspect)	👍 Changes approved	5 changed	Aug 22, 2025, 9:12 AM
v2-vercel (Inspect)	✅ No changes detected	-	Aug 22, 2025, 9:11 AM