Collapse push and pull into one Sync action with the designed dialog

[shaunandrews]

Related issues

• STU-2162
• Stacked on 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 and iterated on this live in the running app; verified with typecheck and the existing unit tests.

Proposed Changes

The header's two arrow buttons (Pull, Push) collapse into one primary "Sync" button that opens a single, designed dialog where everything about a sync is decided:

• Destination — a picker of every connected WordPress.com site (identified by URL, since the Production/Staging label isn't always known), with its last-synced age.
• Direction — a segmented Push / Pull control, with the consequence spelled out ("Replaces the live site with this one." / the reverse).
• What to sync — a file tree that lazily browses the local site (push) or the latest remote backup (pull).

It drives the same push/pull mutations and the same file-tree connector methods #4377 exposes; nothing on the main-process/CLI side changes, and the ported selective-sync dialog is left verbatim — this header opens the designed dialog instead.

Also folds in a header layout fix that surfaced here: the chat header had a leftover flex spacer competing with the toolbar, so the actions landed mid-header instead of pinned right, and both host headers double-padded the toolbar's edge. The toolbar now owns its own edge padding and fills the header.

Testing Instructions

1. Open a connected site → the header shows one Sync button (plus ⋮ for Disconnect).
2. Click Sync → confirm the destination picker (when >1 connection), the Push/Pull segmented control, the consequence line, and the file tree.
3. Flip direction and confirm the tree re-roots (local for push, remote backup for pull) and the selection resets.
4. Run a push and a pull; confirm both behave as before.
5. From onboarding, connect a live site and confirm you land with the dialog open on Pull.
6. Confirm the header actions sit hard-right in both the chat and site-overview headers.
7. 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.