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

[shaunandrews]

Related issues

• STU-2162
• Stacked on Bring back preview links as a Share dialog in the header #4448 (→ Collapse push and pull into one Sync action with the designed dialog #4447 → 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.