This issue was addressed through improved state management. This issue is fixed in Safari 17.6, iOS 17.6 and iPadOS 17.6, macOS Sonoma 14.6. Private Browsing tabs may be accessed without authentication.
https://support.apple.com/kb/HT214121
https://support.apple.com/kb/HT214119
https://support.apple.com/kb/HT214117
https://support.apple.com/en-us/HT214121
https://support.apple.com/en-us/HT214119
https://support.apple.com/en-us/HT214117
https://support.apple.com/en-us/120913
https://support.apple.com/en-us/120911
https://support.apple.com/en-us/120909
https://lists.debian.org/debian-lts-announce/2024/09/msg00006.html