Admin bar: Make frontend search UX similar to prominent command palette

[fushar]

Trac ticket: https://core.trac.wordpress.org/ticket/65221

Summary

This PR updates the frontend search admin bar node to make it similar with the command palette changes from #11796. Now, both the "search" functionalities have more consistent UI.

Desktop

Before

After

Screen.Capture.on.2026-05-12.at.20-29-30.mp4

Mobile

Before	After

Screen.Capture.on.2026-05-12.at.20-35-39.mp4

Alternative considered

I considered making the search input collapsed first as per the current behavior, but move it to the left of the avatar node for more consistency, as follows:

Screen.Capture.on.2026-05-12.at.20-52-10.mp4

But I believe the proposed solution is better.

Use of AI Tools

AI assistance: Yes
Tool(s): Claude Code
Model(s): Claude Opus 4.7
Used for: CSS implementation under my supervision.

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