This issue was addressed through improved state management. This issue is fixed in macOS Sonoma 14.6, iOS 17.6 and iPadOS 17.6, Safari 17.6. Private Browsing tabs may be accessed without authentication.
https://support.apple.com/en-us/HT214121
https://support.apple.com/en-us/HT214119
https://support.apple.com/en-us/HT214117
http://seclists.org/fulldisclosure/2024/Jul/18