[AMPMKT-10009] feat(analytics): instrument docs with GTM and PostHog

[dbarrialitebox]

Description

Docs is the only surface with no analytics of any kind, and it is not being
redesigned, so it gives a reference line the rebuilt pages cannot.

• docs.json — enables GTM container GTM-KKG4NLZD, and adds ?trk=docs to
  the two navbar hrefs so docs can be attributed as a signup source.
• analytics.js (new) — pushes login_button_clicked and cta_clicked to the
  dataLayer, and loads posthog-js. Mintlify includes any .js in the content
  directory on every page.

Element	Event	Properties
Sign in	Login Button Clicked	label · section: navbar · destination_url
Start building now	CTA Clicked	label · section: navbar · element_type: button · conversion_type: signup · destination_url

Two decisions worth knowing:

• The container's tags read the dataLayer, not the DOM, and it does not load
  posthog-js — it assumes an instance is already on the page. Both gaps are what
  analytics.js fills.
• Mintlify's native integrations.posthog is not used. It exposes only
  apiKey / apiHost / sessionRecording, so autocapture cannot be turned off.
  Docs sends two events and nothing else; before_send enforces it.

capture_pageview is off, so docs reports no traffic volume. The two events still
carry $pathname, so no report in the plan breaks.

Open question: consent

The marketing site starts PostHog opted out and waits for
cookieConsent.setConsent(). Docs has no equivalent and this PR adds none. Both
are on *.withampersand.com and share the cookie, so this is one posture, not two.
Needs a call from whoever owns the consent policy before merge.

Screenshot

N/A

[buildstuff-2026]

src/docs.json is the main configuration file for the docs. Do not edit it manually, edit src/generate-docs.ts instead.