Studio: Improve context for errors that bubble up from wordpress-server-child

[katinthehatsite]

Related issues

Fixes STU-1496

How AI was used in this PR

It was used to help identify the issue.

Proposed Changes

This PR ensures that we display more detailed error message to the user from wordpress-server-child

Testing Instructions

• Pull the changes from this branch
• Start Studio with npm start
• Use the broken blueprint e.g.
  broken-blueprint.json
• Click on Add site -> Build a new site -> Upload blueprint
• Observe that you can see more details about why the upload failed:

• Confirm that when you open Studio logs, you can also find more details there:

Pre-merge Checklist

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

[wpmobilebot]

📊 Performance Test Results

Comparing 6183398 vs trunk

app-size

Metric	trunk	6183398	Diff	Change
App Size (Mac)	1409.60 MB	1409.46 MB	0.14 MB	⚪ 0.0%

site-editor

Metric	trunk	6183398	Diff	Change
load	1526 ms	1489 ms	37 ms	⚪ 0.0%

site-startup

Metric	trunk	6183398	Diff	Change
siteCreation	8573 ms	8581 ms	+8 ms	⚪ 0.0%
siteStartup	4927 ms	4931 ms	+4 ms	⚪ 0.0%

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Results are median values from multiple test runs.

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