I have verified this issue using the latest Firefox Release 108.0 (Build ID: 0221208122842) on Windows 10x64, macOS 12.6 and Linux Mint 20.2 and I can confirm that the search bar is still successfully displayed on the "about:privatebrowsing" page.
Bug 1521961 Comment 4 Edit History
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I have verified this issue using the latest Firefox Release 108.0 (Build ID: 20221208122842) on Windows 10x64, macOS 12.6 and Linux Mint 20.2 and I can confirm that the search bar is still successfully displayed on the "about:privatebrowsing" page.