Publish through the redesigned connect-site picker, shared with onboarding

[shaunandrews]

Related issues

• STU-2162
• Stacked on Agentic UI: Add a Learn more modal for publishing and syncing #4448 (→ Agentic UI: Make the site identity a menu with quick actions #4447 → Agentic UI: Replace the site dropdown with a permanent header #4445 → Agentic UI: Port selective sync #4377). Retarget up the chain as each lands.

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

How AI was used in this PR

Claude Code built this on top of the header stack and verified it live (typecheck + unit tests, all passing).

Proposed Changes

The header's Publish action (shown when a site has no live connection yet) now opens the redesigned connect modal — the same rich picker onboarding uses, with search, grouped sites (available / needs transfer / needs upgrade), thumbnails, and per-site status — instead of the plain list it opened before.

To do that without duplicating logic, the picker's site-presentation helpers are promoted into the shared connect-site-picker component that both the Publish modal and onboarding draw from. Onboarding's connect screen itself is unchanged in this PR; migrating its inline picker UI onto the shared component is a sensible follow-up.

Testing Instructions

1. Open a site with no connected WordPress.com site → click Publish.
2. Confirm the modal shows the searchable, grouped site picker (not the old flat list), and connecting a site works.
3. Run onboarding's connect step and confirm it still behaves exactly as before.
4. Both light and dark.

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.

[shaunandrews]

🤖 AI-written review note — verified against the current PR head.

The shared picker still uses onboarding/Pull copy: “Select a site to create its local copy.” In Publish, the user is selecting a remote Push destination.

Also, usePickableWpcomSites() filters out every non-syncable site before the picker receives them, so the “Unavailable” section and transfer/upgrade states implied by this UI cannot appear. Either pass the full site set needed by the grouped picker or remove the unreachable presentation.

Finally, the “Create a WordPress.com site” URL sets autoOpenPush=true, but the existing sync-connect listener ignores that field; returning from checkout connects the site without continuing into Push.