docs: bring rt.cool up to date with rt shipping inside mattstack.app

[m4ttheweric]

rt.cool still described rt as a standalone CLI installed with Homebrew. This audits every hand-written page against the source and fixes what drifted.

What changed

Getting started

• Install is rewritten around mattstack.app: DMG download, the setup window's steps, and the headless rt --post-install path.
• Removes Homebrew, the made-up Chrome Web Store link, and the old PATH states.
• Onboard a repo uses the state.db index and explains opt-in background tracking.
• The intro frames rt as the mattstack CLI and lists the sibling apps.

Guides and partials

• All 16 guides and 11 partials were checked claim by claim against the code, and 22 files changed.
• Biggest fixes: tray (Sparkle updates, real menu), daemon (opt-in tracking, no rt status), state-backup (tools are bundled now), runner (tmux default, missing keys), gates, and teams.

Command tree (generated reference)

• Repo arg hints no longer point at repos.json.
• The waive hint says "waivable", not "finish-gated".
• rt runner defaults to tmux and declares --herdr.

Not in this PR

• rt settings extension looks for the vsix in Contents/MacOS/, but the bundle ships it in Contents/Resources/.
• commands/daemon.ts messages still name repos.json.
• skills/rt-create-plugin/SKILL.md and docs/plugins.md still point at ~/.rt/plugins/.
• The AGENTS.md state-backup note still says the tools come from PATH.

Verification

• bun run docs:check is in sync with 0 coverage gaps.
• The site build passes with links set to throw on break.
• picker:check and the command-tree tests pass.
• Intro and Install were rendered in light and dark.

🤖 Generated with Claude Code

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📥 Commits

Reviewing files that changed from the base of the PR and between 48ff2e4 and 5f4216a.

📒 Files selected for processing (37)

• lib/command-tree-def.ts
• website/docs/getting-started/first-commands.mdx
• website/docs/getting-started/install.mdx
• website/docs/getting-started/onboard-a-repo.mdx
• website/docs/guides/chat.mdx
• website/docs/guides/common-flags.mdx
• website/docs/guides/context-extension.mdx
• website/docs/guides/daemon.mdx
• website/docs/guides/gates.mdx
• website/docs/guides/logging.mdx
• website/docs/guides/mcp.mdx
• website/docs/guides/picker.mdx
• website/docs/guides/plugins.mdx
• website/docs/guides/proxy.mdx
• website/docs/guides/runner.mdx
• website/docs/guides/state-backup.mdx
• website/docs/guides/strongdm.mdx
• website/docs/guides/teams-and-invites.mdx
• website/docs/guides/tray.mdx
• website/docs/intro.mdx
• website/docs/reference/_partials/agent.mdx
• website/docs/reference/_partials/mr/map.mdx
• website/docs/reference/_partials/runs/run-start.mdx
• website/docs/reference/_partials/setup/waive.mdx
• website/docs/reference/_partials/state/backup/index.mdx
• website/docs/reference/_partials/state/restore.mdx
• website/docs/reference/daemon/track.mdx
• website/docs/reference/global.mdx
• website/docs/reference/runner.mdx
• website/docs/reference/settings/explain.mdx
• website/docs/reference/settings/get.mdx
• website/docs/reference/settings/list.mdx
• website/docs/reference/settings/set.mdx
• website/docs/reference/settings/unset.mdx
• website/docs/reference/setup/unwaive.mdx
• website/docs/reference/setup/waive.mdx
• website/docusaurus.config.ts

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