Replace the site dropdown with a permanent header

[shaunandrews]

Related issues

• STU-2162
• Stacked on Agentic UI: Port selective sync #4377 (agenticPortSelectiveSync). This PR targets that branch, not trunk, and reuses the selective-sync dialog it introduces. Retarget to trunk once Agentic UI: Port selective sync #4377 lands.

⚠️ Visual change: needs human review in light + dark mode.

How AI was used in this PR

Claude Code reconstructed this header from a larger exploration branch on top of #4377, keeping it to the smallest change that replaces the dropdown, and verified it in isolation (typecheck + unit tests).

Proposed Changes

The site dropdown is gone. Its actions were hidden behind a trigger that also had to read as a status indicator — you couldn't tell it was a menu, and everything the site could do lived one unlabelled click away.

In its place is a permanent site header: the site's icon, name, and local URL on the left; its actions as real, visible buttons on the right. What used to be buried in the menu is now on the surface:

• Pull and Push open the selective-sync dialog (the one Agentic UI: Port selective sync #4377 ports from the classic UI).
• Publish connects a WordPress.com site when none is connected yet.
• Disconnect lives in an overflow next to the sync actions.
• Log in replaces the actions when signed out.
• Start/stop and the run-state dot stay with the site's identity (the dot only appears when the sidebar — which already shows one per site — is collapsed).

This is the lean first step of a larger header effort. It deliberately keeps the existing selective-sync dialog and separate Pull/Push buttons; consolidating those into a single Sync action, and bringing preview sharing back as a Share button, are follow-up PRs on top of this one.

Known gap: preview links are not reachable in this PR — that surface moves to the upcoming Share dialog rather than staying in a dropdown that's going away.

Testing Instructions

1. Open a site in the agentic UI. Confirm the header shows the site icon, name, and (when running) a clickable local URL, with action buttons on the right.
2. Not connected: the primary action is Publish — it opens the site picker; connecting a WordPress.com site works as before.
3. Connected: Pull and Push each open the selective-sync dialog; a sync runs and reports as before. The overflow (⋮) offers Disconnect.
4. Signed out: the header shows Log in, which opens the login flow.
5. Collapse the sidebar and confirm the run-state dot appears in the header; expand it and confirm the dot is only in the sidebar rows.
6. From onboarding, connect a live site and confirm you land on the site with the Pull dialog opened automatically.
7. Both light and dark appearance.

Pre-merge Checklist

• I have considered if this change warrants user-facing release notes and have added them to RELEASE-NOTES.txt if necessary.
• I have added or updated tests to cover my changes, if applicable.
• I have tested my changes in both light and dark mode.