Agentic UI: Add default site directory preference to settings

[bcotrim]

Related issues

• Related to STU-1882

How AI was used in this PR

Implemented with Claude Code following the auto-save settings patterns from #4219, using the Save-button-era exploration in #3979 as a reference. All code reviewed, linted, typechecked, and unit-tested locally.

Proposed Changes

• A new Default site directory row in Settings shows the current folder and opens the native folder picker; picking a folder saves immediately, matching the auto-save behavior of the other preferences. The value is shared with the classic UI, so changes made in either stay in sync.

Light	Dark

Testing Instructions

1. Enable the new Studio experience and open Settings.
2. Verify the Default site directory row shows your current default (or "Choose a folder…" if unset).
3. Click it, pick a folder — the choice should persist immediately (no Save button), and survive an app restart.
4. Cancel the picker — nothing should change or save.
5. Switch to the classic UI → Preferences: the same directory should appear there (and vice versa).
6. Create a new site in the agentic UI: the proposed path should live under the chosen directory.
7. Check the settings screen in both light and dark mode.

Pre-merge Checklist

• Have you checked for TypeScript, React or other console errors?

🤖 Generated with Claude Code

[wpmobilebot]

📊 Performance Test Results

Comparing d6a2b3a vs trunk

app-size

Metric	trunk	d6a2b3a	Diff	Change
App Size (Mac)	1361.84 MB	1355.67 MB	6.17 MB	🟢 -0.5%

site-editor

Metric	trunk	d6a2b3a	Diff	Change
load	1047 ms	1052 ms	+5 ms	⚪ 0.0%

site-startup

Metric	trunk	d6a2b3a	Diff	Change
siteCreation	6998 ms	7013 ms	+15 ms	⚪ 0.0%
siteStartup	2353 ms	2355 ms	+2 ms	⚪ 0.0%

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Results are median values from multiple test runs.

Legend: 🟢 Improvement (faster) | 🔴 Regression (slower) | ⚪ No change (<50ms diff)

[nightnei]

Works perfectly. I also tested with spece in the site folder (as temporary studio) and confirm that it works w/o issues.
The code LGTM.