Agentic UI: Add Connections to site overview

[shaunandrews]

Related issues

• Related to Design Exploration: Rebuild the site overview around summary cards #4462

How AI was used in this PR

AI helped extract the Connections slice from the larger overview exploration, adapt it to the current branch stack, and build coverage for its user states and actions. I reviewed the diff and the rendered result.

Proposed Changes

This is the first PR in the site-management UI stack:

1. Agentic UI: Add Connections to site overview #4474 — Connections on the overview (this PR)
2. Agentic UI: Add preview sites to site overview #4483 — Preview sites on the overview
3. Design Exploration: Replace the site dropdown with a permanent site header #4400 — Persistent site header with Share and Sync dialogs
4. Agentic UI: Integrate the site overview, sharing, and sync progress #4601 — Integrated overview polish, reusable sharing controls, and operation progress

The goal of this layer is to make a local site's live-site relationships visible without leaving Overview.

• Show every live site connected to the selected local site.
• Show the most recent push or pull time and distinguish Production from Staging.
• Provide connect, sync, open, and copy entry points.
• Cover signed-out, disconnected, connected, stale, and in-progress states.

The later stack layers refine these actions into the final Push/Pull layout, open the shared Sync dialog, and add inline plus persistent progress feedback.

Current UI

Testing Instructions

1. Open a local site's Overview.
2. Confirm Connections appears as its own card below About.
3. Verify a connected site shows its URL, last sync, and environment.
4. Exercise the connect, sync, open, and copy actions.
5. Check signed-out, disconnected, loading, light, and dark states.

Verification used while preparing this stack:

• Built this exact PR head with the local browser UI target.
• Rendered it against the local Studio backend and captured the screenshot above.
• The focused overview tests and full typecheck pass on the integrated stack head.

Pre-merge Checklist

• Have you checked for TypeScript, React or other console errors?