Studio: Add search to current library

[katinthehatsite]

Related issues

Related to STU-1634

How AI was used in this PR

It was used to implement the feature.

Proposed Changes

This PR ensures that we add the search feature to the current blueprints library that is pulled from the public Blueprints Library through the API.

Testing Instructions

• Pull the changes from this branch
• Start the app with npm start
• Enable the Blueprints Gallery Library feature flag
• Click on Add site > Build new site
• Scroll down to the blueprints library
• Confirm that you can use the search box and search by title or category

Pre-merge Checklist

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

[wpmobilebot]

📊 Performance Test Results

Comparing fe9894d vs trunk

app-size

Metric	trunk	fe9894d	Diff	Change
App Size (Mac)	1410.41 MB	1410.41 MB	+0.00 MB	⚪ 0.0%

site-editor

Metric	trunk	fe9894d	Diff	Change
load	1511 ms	1498 ms	13 ms	⚪ 0.0%

site-startup

Metric	trunk	fe9894d	Diff	Change
siteCreation	8104 ms	8090 ms	14 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)