Replace the ported sync dialog with the designed one from the original sketch

[shaunandrews]

Related issues

• STU-2162
• Stacked on Publish through the redesigned connect-site picker, shared with onboarding #4449 (→ 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).

⚠️ Visual change: needs human review in light + dark mode.
🧹 Stack note: this supersedes the interim approach in #4447 (a direction toggle bolted onto the ported classic dialog). Before the stack goes ready-for-review it will be folded down into #4447, and #4447's edits to the ported selective-sync dialog dropped, so the classic dialog stays untouched.

How AI was used in this PR

Claude Code built this on top of the header stack and iterated on it live in the running app; verified with typecheck.

Proposed Changes

Replaces the classic selective-sync dialog (ported verbatim by #4377, styled for the old renderer) with the designed sync dialog from the original sketch, so the Sync surface matches the rest of the agentic UI — the header, the Share dialog, everything else.

One place answers everything a sync needs:

• 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), so a sync can carry exactly what's selected.

It drives the same push/pull mutations and the same file-tree connector methods #4377 already exposes; nothing on the main-process/CLI side changes.

Testing Instructions

1. Open a connected site → Sync.
2. Confirm the destination picker (when there's more than one connection), the Push/Pull segmented control, the consequence line, and the file tree all render in the agentic style.
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 and confirm both behave as before.
5. From onboarding, connect a live site and confirm you land with the dialog open on Pull.
6. 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]

Folded into #4447 — the designed sync dialog now lands there directly, and #4447 no longer modifies the ported selective-sync dialog (it stays verbatim). Closing this in favor of the cleaner stack.